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THEOEY OF GEOUPS 
 
 OF 
 
 FINITE OEDEF. 
 
aonDon: C. J. CLAY and SONS, 
 
 GAMBEIDGE UNIVEBSITY PEESS WAREHOUSE, 
 
 AVE MARIA LANE. 
 
 Blmenia: 263, ARGILE STREET. 
 
 Hcipjig: F. A. BROCKHAUS. 
 j^tto Hotft: THE MACMILLAN COMPANY. 
 
THEOEY OF GROUPS 
 
 OF 
 
 FINITE OEDEE 
 
 BY 
 
 W. BURNSIDE, M.A., F.R.S., 
 
 LATE FELLOW OF PEMBROKE COLLEOE, CAMBRIDGE ; 
 PROFESSOR OF MATHEMATICS AT THE ROYAL NAVAL COLLEGE, GREENWICH. 
 
 OAMBKIDGE : 
 AT THE UNIVERSITY PRESS. i 
 
 1897 
 
 [All Rights reserved.] ' \ M '' 
 
dambtitip: 
 
 PRINTED BY J. AND 0. f. OLAT, 
 AT THE BNIVEBSITT PKESS. 
 
PREFACE. 
 
 THE theory of groups of finite order may be said to date 
 from the time of Cauchy. To him are due the first 
 attempts at classification with a view to forming a theory from 
 a number of isolated facts. Galois introduced into the theory 
 the exceedingly important idea of a self-conjugate sub-group, 
 and the corresponding division of groups into simple and com- 
 posite. Moreover, by shewing that to every equation of finite 
 degree there corresponds a group of finite order on which 
 all the properties of the equation depend, Galois indicated 
 how far reaching the applications of the theory might be, and 
 thereby contributed greatly, if indirectly, to its subsequent 
 developement. 
 
 Many additions were made, mainly by J'rench mathe- 
 maticians, during the middle part of the century. The first 
 connected exposition of the theory was given in the third 
 edition of M. Serret's " Gours d'Algebre Supdrieure,'' which was 
 published in 1866. This was followed in 1870 by M. Jordan's 
 "TraiU des substitutions et des Equations algihriques." The 
 greater part of M. Jordan's treatise is devoted to a develope- 
 ment of the ideas of Galois and to their application to the 
 theory of equations. 
 
 No' considerable progress in the theory, as apart from its 
 applications, was made till the appearance in 1872 o^ Herr 
 Sylow's memoir " TMorhmes sur les groupes de substitutions" 
 in the fifth volume of the Mathematische Annalen. Since the 
 date of this memoir, but more especially in recent years, the 
 theory has advanced continuously. 
 
 In 1882 appeared Herr Netto's " Substitutionentheorie und 
 
VI PREFACE. 
 
 ihre Anwendungen auf die Algebra^' in which, as in M. Serret's 
 and M. Jordan's works, the subject is treated entirely from the 
 point of view of groups of substitutions. Last but not least 
 among the works which give a detailed account of the subject 
 must be mentioned Herr Weber's " Lehrbuch der Algebra," of 
 which the first volume appeared in 189-5 and the second in 
 1896. In the last section of the first volume some of the more 
 important properties of substitution groups are given. In the 
 first section of the second volume, however, the subject is 
 approached from a more general point of view, and a theory of 
 finite groups is developed which is quite independent of any 
 special mode of representing them. 
 
 The present treatise is intended to introduce to the reader 
 the main outlines of the theory of groups of finite order apart 
 from any applications. The subject is one which has hitherto 
 attracted but little attention in this country ; it will afford 
 me much satisfaction if, by means of this book, I shall succeed 
 in arousing interest among English mathematicians in a branch 
 of pure mathematics which becomes the more fascinating the' 
 more it is studied. 
 
 Oayley's dictum that " a group is defined by means of the 
 laws of combination of its symbols" would imply that, in dealing 
 purely with th^ theory of groups, no more concrete mode of 
 representation should be used than is absolutely necessary. 
 It may then be asked why, in a book which professes to leave 
 all applications on one side, a considerable space is devoted to 
 substitution groups; while other particular modes of repre- 
 sentation, such as groups of linear transformations, are not 
 even referred to. My answer to this question is that while, in 
 the present state of our knowledge, many results in the pure 
 theory are arrived at most readily by dealing with properties 
 of substitution groups, it would be difiicult to find a result that 
 could be most directly obtained by the consideration of groups 
 of linear transformations. 
 
 The plan of the book is as follows. The first Chapter has 
 been devoted to explaining the notation of substitutions. As 
 this notation may not improbably be unfamiliar to many 
 English readers, some such introduction is necessary to make 
 
PREFACE. vii 
 
 the illustrations used in the following chapters intelligible. 
 Chapters II to VII deal with the more important properties of 
 groups which are independent of any special form of repre- 
 sentation. The notation and methods of substitution groups 
 have been rigorously excluded in the proofs and investigations 
 contained in these chapters ; for the purposes of illustration, 
 however, the notation has been used whenever convenient. 
 Chapters VIII to X deal with those properties of groups which 
 depend on their representation as substitution groups. Chapter 
 XI treats of the isomorphism of a group with itself. Here, 
 though the properties involved are independent of the form of 
 representation of the group, the methods of substitution groups 
 are partially employed. Graphical modes of representing a 
 group are considered in Chapters XII and XIII. In Chapter 
 XIV the properties of a class of groups, of great importance in 
 analysis, are investigated as a general illustration of the 
 foregoing theory. The last Chapter contains a series of results 
 in connection with the classification of groups as simple, 
 composite, or soluble. 
 
 A few illustrative examples have been given throughout 
 the book. As far as possible I have selected such examples 
 as would serve to complete or continue the discussion in the 
 text where they occur. 
 
 In addition to the works by Serret, Jordan, Netto and 
 Weber already referred to, I have while writing this book 
 consulted many original memoirs. Of these I may specially 
 mention, as having been of great use to me, two by Herr Dyck 
 published in the twentieth and twenty-second volumes of the 
 Mathematische Annalen with the title " Oruppentheoretische 
 Studien " ; three by Herr Frobenius in the Berliner Sitzungs- 
 berichte for 1895 with the titles, " Ueber endliche Qruppen," 
 " Ueber auflosbare Oruppen," and " Verallgemeinerung des 
 Sylow'scken Satzes " ; and one by Herr Holder in the forty- 
 sixth volume of the Mathematische Annalen with the title 
 " Bildwng zusamm&ngesetzter Oruppen." Whenever a result 
 is taken from an original memoir I have given a full reference ; 
 any omission to do so that may possibly occur is due to an 
 oversight on my part. 
 
Vlll PREFACE. 
 
 To Mr A. R. Forsyth, Sc.D., F.R.S., Fellow of Trinity 
 College, Cambridge, and Sadlerian Professor of Mathematics, 
 and to Mr G.B. Mathews, M.A., F.R.S., late Fellow of St John's 
 College, Cambridge, and formerly Professor of Mathematics in 
 the University of North Wales, I am under a debt of gratitude 
 for the care and patience with which they have read the proof- 
 sheets. Without the assistance they have so generously 
 given me, the errors and obscurities, which I can hardly hope 
 to have entirely escaped, would have been far more numerous. 
 I wish to express my grateful thanks also to Prof 0. Holder 
 of Konigsberg who very kindly read and criticized parts of 
 the last chapter. Finally I must thank the Syndics of the 
 University Press of Cambridge for the assistance they have 
 rendered in the publication of the book, and the whole Staff 
 of the Press for the painstaking and careful way in which 
 the printing has been done. 
 
 W. BURNSIDE. 
 
 July, 1897. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 ON SUBSTITUTIONS. 
 
 §§ PAGE 
 
 1 Object of the chapter 1 
 
 2 Definition of a substitution 1 
 
 3 — 6 Notation for substitutions ; cycles ; . products of substitu- 
 tions 1 — 4 
 
 7, 8 Identical substitution ; inverse substitutions ; order of a 
 
 substitution 4 — 6 
 
 9, 10 Circular, regular, similar, and permutable substitutions . 7, 8 
 11 Transpositions ; representation of a substitution as a product 
 
 of transpositions ; odd and even substitutions . . 9, 10 
 
 CHAPTER II. 
 
 THE DEFINITION OF A GROUP. 
 
 12 Definition of a group 11, 12 
 
 13 Identical operation 12, 13 
 
 14 Continuous, discontinuous, and mixed groups . . .13, 14 
 15, 16 Order of an operation ; products of operations . . . 14 — 16 
 
 17 Examples of groups of operations; multiplication table of a 
 
 group .......... 17 — 20 
 
 18, 19 Generating operations of a group; defining relations; simply 
 
 isomorphic groups 20 — 22 
 
 20 Dyck's theorem 22—24 
 
 21 Various modes of representing groups 24 
 
CONTENTS. 
 
 CHAPTER III. 
 
 ON THE SIMPLER PROPERTIES OF A GROUP WHICH ARE 
 INDEPENDENT OF ITS MODE OF REPRESENTATION. 
 
 22, 23 Sub-groups ; the order of a sub-group divides the order of 
 
 the group containing it; symbol for a group . . 25 — 27 
 24 Transforming one operation by another; conjugate opera- 
 tions and sub-groups ; self-conjugate operations and 
 sub-groups ; simple and composite groups . . . 27 — 29 
 25, 26 The operations of a group which are permutable with a 
 given operation or sub-group form a group ; complete 
 sets of conjugate operations or sub-groups . . . 29 — 33 
 27 Theorems on self-conjugate sub-groups; maximum sub- 
 groups ; maximum self-conjugate sub-groups . . 33 — 35 
 28 — 31 Multiply isomorphic groups; factor groups; direct product 
 
 of two groups 35 — iO 
 
 32 General isomorphism between two groups . . . . 40, 41 
 
 33 — 35 Permutable groups; Examples 41 — 45 
 
 CHAPTER IV. 
 
 ON ABELIAN GROUPS. 
 
 36 — 88 Sub-groups of Abelian groups ; every Abelian group is the 
 direct product of Abelian groups whose orders are 
 
 powers of different primes 46 — 48 
 
 39 Limitation of the discussion to Abelian groups whose orders 
 
 are powers of primes 48 
 
 40 — 44 Existence of a set of independent generating operations of 
 such a group ; invariauce of the orders of the gene- 
 rating operations; symbol for Abelian group of given 
 
 type 49 — 55 
 
 45 — 47 Determination of all types of sub-groups of a given Abelian 
 
 group 55 — 58 
 
 48, 49 Properties of an Abelian group of type (1, 1, ..., 1) . . 58 — 60 
 
 50 Examples 60 
 
CONTENTS. xi 
 
 CHAPTER V. 
 
 ON GROUPS WHOSE ORDERS ARE POWERS OF PRIMES. 
 
 §§ PAOE 
 
 51 Object of the chapter 61 
 
 52 Every group whose order is the power of a prime contains 
 
 self-conjugate operations 61, 62 
 
 53 — 58 General properties of groups whose orders are powers of 
 
 primes 62—69 
 
 59 — 61 The number of sub-groups of order p' of a group of order 
 
 p^, where p is a, prime, is congruent to unity, 
 
 mod. p • . . . . 69—71 
 
 62, 63 Groups of order p"^ with a single sub-group of order p' . 71 — 75 
 64 — 67 Groups of order p™ with a self-conjugate cyclical sub-group 
 
 of order ^™-2 75—81 
 
 68 Distinct types of groups of orders p'' and p^ . . . 81, 82 
 
 69 — 72 Distinct types of groups of order p* 82 — 86 
 
 73, 74 Tables of groups of orders p'^, p^, and p* ■ • • . 86 — 89 
 
 75 Examples 89 
 
 CHAPTER VI. 
 
 ON SYLOW'S THEOREM. 
 
 76 Object of the chapter 90, 91 
 
 77, 78 Proof of Sylow's theorem 91—94 
 
 79 — 82 Direct consequences of Sylow's theorem . . . 95 — 100 
 83 — 85 Distinct types of groups of orders pq (p and j being 
 
 different primes), 24, and 60 100—108 
 
 86 Generalization of Sylow's theorem .... 108 — 110 
 
 87 Frobenius's theorem 110 — 115 
 
 88 Groups with properties analogous to those of groups 
 
 whose orders are powers of primes . . . 115 — 117 
 
 CHAPTER VII. 
 
 ON THE COMPOSITION-SERIES OF A GROUP. 
 
 89 The composition-series, composition-factors, and factor- 
 groups of a given group 118, 119 
 
 90, 91 Invarianoe of the composition-series of a group . - . 119—122 
 
Xll CONTENTS. 
 
 §§ VAOE 
 
 92 The chief composition-series, or chief series of a group ; 
 
 its invariance; construction of a composition- series 
 
 from a chief series 122, 123 
 
 93—95 Types of the factor-groups of a chief series; minimum 
 
 self-conjugate sub-groups 124 — 127 
 
 96, 97 Examples of composition-series 128, 129 
 
 98-100 Soluble groups 129—132 
 
 101 Distinct types of groups of order p^q, where p and q 
 
 are different primes 132 — 137 
 
 CHAPTER VIII. 
 
 ON SUBSTITUTION GROUPS : TRANSITIVE AND INTRANSITIVE 
 
 GROUPS. 
 
 102 Substitution groups; degree of a group . . . 138, 189 
 
 103 The symmetric and the alternating groups . . . 139, 140 
 
 104 Transitive and intransitive groups; the degree of a 
 
 transitive group is a factor of its order . . 140, 141 
 
 105 Transitive groups whose substitutions displace all or 
 
 all but one of the symbols 141 — 144 
 
 106, 107 Self-conjugate operations and sub-groups of transitive 
 groups ; transitive groups of which the order is 
 
 equal to the degree 144 — 148 
 
 108, 109 Multiply transitive groups; the order of a ft-ply transi- 
 tive group of degree n is divisible by n{n~l)... 
 (re - * -I- 1) ; construction of multiply transitive 
 groups 148 — 151 
 
 110 Groups of degree n, which do not contain the alter- 
 
 nating group, cannot be more than (Jn + l)-ply 
 transitive 151 — 153 
 
 111 The alternating group of degree n is simple, except 
 
 when n is 4 153, 154 
 
 112, 113 Examples of doubly and triply transitive groups . . 154 — 158 
 
 114 — 116 Properties of intransitive groups 159 — 162 
 
 117 Intransitive groups of degree 7 163, 164 
 
 118, 119 Number of symbols left unchanged by all the substi- 
 tutions of a group is the product of the order of 
 the group and the number of the sets in which 
 the symbols are interchanged transitively . . 165 — 167 
 Notes to §§ 108, 110 167—170 
 
CONTENTS. xiii 
 
 CHAPTER IX. 
 
 ON SUBSTITUTION GROUPS: PRIMITIVE AND IMPRIMITIVE 
 GROUPS. 
 
 §§ PA8E 
 
 120 Object of the chapter I7I 
 
 121 Imprimitive and primitive groups; imprimitive systems 171, 172 
 122 — 125 Eepresentation of any group in transitive form; primi- 
 
 tivity or imprimitivity of the group so represented 172 — 179 
 
 126 Number of distinct modes of representing the alternat- 
 
 ing group of degree 5 in transitive form . . 179, 180 
 
 127 Imprimitive groups of degree 6 180 — 183 
 
 128, 129 Tests of primitivity: properties of imprimitive systems 183 — 186 
 
 130 Self-conjugate sub-groups of transitive groups ; a self- 
 
 conjugate sub-group of a primitive group must be 
 
 transitive 186—188 
 
 131 Self-conjugate sub-groups of ft-ply transitive groups are 
 
 in general (k - l)-ply transitive .... 188, 189 
 132 — 136 Further properties of self -conjugate sub-groups of primi- 
 tive groups 190 — 194 
 
 137 Examples .... .... 194, 195 
 
 CHAPTER X. 
 
 on substitution groups : transitivity and primitivity : 
 (concluding properties). 
 
 138 References to tables of primitive groups . . . 196 
 139 — 141 Primitive groups with transitive sub-groups of smaller 
 degree: limit to the order of a primitive group of 
 
 given degree 196 — 199 
 
 142 Properties of the symmetric group .... 199, 200 
 143 — 145 Further limitations on the orders of primitive groups 
 
 of given degree 200 — 204 
 
 146 Primitive groups whose degrees do not exceed 8 . . 205—211 
 147 — 149 Sub-groups of doubly transitive groups which leave two 
 
 symbols unchanged; complete sets of triplets . 212 — 215 
 150, 151 The most general groups each of whose substitutions 
 is permutable with a given substitution, or with 
 
 every substitution of a given group . . . 215 — 217 
 
 152 Transitive groups whose orders are powers of primes . 218, 219 
 
 153 Example 220 
 
XIV 
 
 CONTENTS. 
 
 CHAPTER XI. 
 
 ON THE ISOMORPHISM OF A GROUP WITH ITSELF. 
 
 §§ PAGE 
 
 154 Object of the chapter 221 
 
 155, 156 Isomorphism of a group with itself; the group of iso- 
 morphisms 222—224 
 
 157 Cogredient and contragredient isomorphisms; the group 
 
 of cogredient isomorphisms is contained self-con- 
 
 jugately in the group of isomorphisms . . . 224 — 226 
 
 158 The holomorph of a group 226—228 
 
 159 — 161 Properties of isomorphisms ; representation of the group 
 
 of isomorphisms in transitive form . . . 228 — 232 
 162 Characteristic sub-groups; groups with no character- 
 istic sub-groups 232 
 
 163, 164 Characteristic series of a group ; its invariance ; charac- 
 teristic series of a group whose order is the power 
 of a prime 232—235 
 
 165—167 Complete groups 235—239 
 
 168 — 170 The group of isomorphisms and the holomorph of a 
 
 cyclical group 239—242 
 
 171, 172 The group of isomorphisms and the holomorph of an 
 Abelian group of order p" and type (1, 1, ..., 1); 
 the homogeneous linear group .... 242 — 245 
 
 173 The group of isomorphisms of the alternating group . 245, 246 
 
 174 The group of isomorphisms of doubly transitive groups 
 
 of degree J)" -t-1 and order 4p"(2)'^''-1) . . . 246—249 
 175 — 178 Further properties of isomorphisms ; the symbols 3- (P) 
 
 and e (P) 249—253 
 
 179 Examples 258, 254 
 
 CHAPTER XII. 
 
 ON THE GRAPHICAL REPRESENTATION OF A GROUP. 
 
 180 
 181, 182 
 
 183 
 184—187 
 188—190 
 191—195 
 
 196 
 197, 198 
 
 Groups with an infinite number of operations 
 The most general discontinuous group that can be 
 generated by a finite number of operations; rela 
 tion of special groups to the general group . 
 Graphical representation of a cyclical group 
 Graphical representation of a general group 
 Graphical representation of a special group 
 Graphical representation of groups of finite order 
 
 The genus of a group 
 
 Limitation on the order and on the number of defining 
 
 relations of a group of given genus 
 Note to § 194 
 
 255 
 
 256—259 
 260-262 
 262—267 
 267—272 
 272—280 
 280 
 
 281—284 
 284 
 
CONTENTS. XV 
 
 CHAPTER XIII. 
 
 ON THE GRAPHICAL REPRESENTATION OF GROUPS: GROUPS OF 
 GENUS ZERO AND UNITY: CAYLEY's COLOUR GROUPS. 
 
 §§ PAOE 
 
 199 — 203 Groups of genus zero; their defining relations and 
 
 graphical representation ' . 285 — 292 
 
 204 — 209 Groups of genus unity; their defining relations and 
 
 graphical representation 293 — 302 
 
 210 The graphical representation and the defining relations 
 
 of the simple group of order 168 . . . . 302—305 
 
 211—214 Cayley's colour groups 306—310 
 
 CHAPTER XIV. 
 
 ON THE LINEAR GROUP. 
 
 215 The homogeneous linear group 311, 312 
 
 216—220 Its composition-series 312—817 
 
 221 The simple group which it defines .... 317—319 
 
 222 — 234 The fractional linear group ; determination of the orders 
 of its operations and of their distribution in con- 
 jugate sets; determination of aU of its sub-groups 
 and of their distribution in conjugate sets ; its 
 
 representation as a doubly transitive group . . 319 — 334 
 
 235 Generalization of the fractional linear group . . 334, 335 
 
 236 — 238 Representation of the simple group, defined by the 
 homogeneous linear group, as a doubly transitive 
 
 group; special cases 336 — 340 
 
 239, 240 Generalization of the homogeneous linear group . . 340 — 342 
 
 CHAPTER XV. 
 
 ON SOLUBLE AND COMPOSITE GROUPS. 
 
 241 Object of the chapter ....:.. 343, 344 
 
 242 Direct applications of Sylow's theorem often shew that 
 
 a group of given order must be composite . . 344, 345 
 243—245 Soluble groups whose orders are p"g^, where p and q 
 
 are primes ........ 345 — 352 
 
XVI CONTENTS. 
 
 §§ PAGE 
 
 246 Groups whose sub-groups of order p°- are all eyclioal, 
 
 p"^ being any power of a prime which divides the 
 
 order 352, 353 
 
 247 Groups whose orders contain no square factor . . 353, 354 
 248, 249 Further tests of solubility ; groups whose orders con- 
 tain no cube factor 354 — 360 
 
 250 — 257 Groups of even order in which the operations of odd 
 order form a self-conjugate sub-group; either 12, 
 16, or 56 must divide the order of a simple group 
 
 if it is even 360—366 
 
 258 The simple groups whose orders contain less than 6 
 
 prime factors 367—370 
 
 259, 260 The simple groups whose orders do not exceed 660 . 370 — 375 
 
 261 — 263 Non-soluble composite groups 375 — 378 
 
 Notes to §§ 257, 258, 260 379 
 
 APPENDIX: On French and German technical terms . . . 380—382 
 
 INDEX 383-388 
 
 EEEATA. 
 
 p. 60, line 18, for 7 read 56. 
 
 p. 68, line 10, after p^' add where p is an odd prime. 
 
 p. 150, line 18, after G add which displaces a^. 
 
 p. 150, line 26, after S, add either ASA belongs to G or. 
 
 p. 151, line 6, after form add The reduction is here carried out 
 
 on the supposition that S, and S, displace Oj. 
 
 The modification, when this is not the case, is 
 
 obvious. 
 
CHAPTEE I. 
 
 ON SUBSTITUTIONS. 
 
 1. Among the various notations used in the following 
 pages, there is one of such frequent recurrence that a certain 
 readiness in its use is very desirable in dealing with the 
 subject of this treatise. We therefore propose to devote a 
 preliminary chapter to explaining it in some detail. 
 
 2. Let (Xi, a^,..., an be a set of n distinct letters. The 
 operation of replacing each letter of the set by another, which 
 may be the same letter or a different one, when carried out 
 under the condition that no two letters are replaced by one and 
 the same letter, is called a sxibstitution performed on the n 
 letters. Such a substitution will change any given arrange- 
 ment 
 
 <X\, 0,^,.., , fln 
 
 of the n letters into a definite new arrangement 
 
 6l, &2,..., &„ 
 
 of the same n letters. 
 
 3. One obvious form in which to write the substitution is 
 
 /Oti, Oii, , (ln\ 
 
 V6i, &!, , 6n/ 
 
 thereby indicating that each letter in the upper line is to be 
 replaced by the letter standing under it in the lower. The 
 disadvantage of this form is its unnecessary complexity, each 
 
 B. 1 
 
 7a 
 
2 NOTATION FOR REPRESENTING [3 
 
 of the n letters occurring twice in the expression for the 
 substitution; by the following process, the expression of the 
 substitution may be materially simplified. 
 
 Let p be any one of the n letters, and q the letter in the 
 lower line standing under p in the upper. Suppose now that r 
 is the letter in the lower line that stands under q in the upper, 
 and so on. Since the number of letters is finite, we must arrive 
 at last at a letter s in the upper line under which p stands. 
 If the set of n letters is not thus exhausted, take any letter p' 
 in the upper line, which has not yet occurred, and let j', r',... 
 follow it as q,r,... followed p, till we arrive at s' in the upper 
 line with p' standing under it. If the set of n letters is still not 
 exhausted, repeat the process, starting with a letter p" which 
 has not yet occurred. Since the number of letters is finite, we 
 must in this way at last exhaust them ; and the n letters are 
 thus distributed into a number of sets 
 
 p, q, r,...,s ; 
 
 p', q', r',...,s' ; 
 
 p",q",r",...,s"; 
 
 such that the substitution replaces each letter of a set by the 
 one following it in that set, the last letter of each set being re- 
 placed by the first of the same set. 
 
 If now we represent by the symbol 
 
 {pqr...s) 
 
 the operation of replacing p by q, q by r,..., and s by p, the 
 substitution will be completely represented by the symbol 
 
 {pqr. . .s) ( p'qV. . .s') (p'Yr". ■ -s") 
 
 The advantage of this mode of expressing the substitution is 
 that each of the letters occurs only once in the symbol. 
 
 4. The separate components of the above symbol, such as 
 {pqr...s) are called the cycles of the substitution. In particular 
 cases, one or more of the cycles may contain a single letter ; 
 when this happens, the letters so occurring singly are unaltered 
 by the substitution. The brackets enclosing single letters may 
 
6] SUBSTITUTIONS 3 
 
 clearly be omitted without risk of ambiguity, as also may the 
 unaltered letters themselves. Thus the substitution 
 
 (a, b, c, d, e\ 
 
 fa, b, c, d, e\ 
 \c, b, d, a, el 
 
 may be written (acd) (b) (e), or (acd) be; or simply (acd). If for 
 any reason it were desirable to indicate that substitutions of the 
 five letters a, b, c, d, e were under consideration, the second of 
 these three forms would be used. 
 
 5. The form thus obtained for a substitution is not unique. 
 The symbol (qr...sp) clearly represents the same substitution 
 as (pqr. . .s), if the letters that occur between r and s in the two 
 symbols are the same and occur in the same order ; so that, as 
 regards the letters inside the bracket, any one may be chosen 
 to stand first so long as the cyclical order is preserved un- 
 changed. 
 
 Moreover the order in which the brackets are arranged is 
 clearly immaterial, since the operation denoted by any one 
 bracket has no effect on the letters contained in the other 
 brackets. This latter property is characteristic of the par- 
 ticular expression that has been obtained for a substitution; 
 it depends upon the fact that the expression contains each 
 of the letters once only. 
 
 6. When we proceed to consider the effect of performing 
 two or more substitutions successively, it is seen at once that 
 the order in which the substitutions are carried out in general 
 affects the result. Thus to give a very simple instance, the 
 substitution {ab) followed by (ac) changes a into b, since b is 
 unaltered by the second substitution. Again, (ab) changes b 
 into a and (ac) changes a into c, so that the two substitutions 
 performed successively change b into c. Lastly, (ab) does not 
 affect c and (ac) changes c into a. Hence the two substitutions 
 performed successively change a into b, b into c, c into a, and 
 affect no other symbols. The result of the two substitutions 
 performed successively is therefore equivalent to the substitu- 
 tion (abc) ; and it may be similarly shewn that (ac) followed 
 by (ab) gives (acb) as the resulting substitution. To avoid 
 ambiguity it is therefore necessary to assign, once for all, the 
 
 1—2 
 
4 POWERS AND PRODUCTS [6 
 
 meaning to be attached to such a symbol as SiS^, where s^ and 
 Sj are the symbols of two given substitutions. We shall always 
 understand by the symbol SiS^ the result of carrying out first 
 the substitution Sj and then the substitution Sa- Thus the two 
 simple examples given above may be expressed in the form 
 
 (ab) (ac) = {abc), 
 
 (ac) (ab) = (acb), 
 
 the sign of equality being used to represent that the substitu- 
 tions are equivalent to each other. 
 
 If now 
 
 Sih = Si and 3253 = 85, 
 
 the symbol SjS^Sg may be regarded as the substitution s^ followed 
 by «3 or as s^ followed by Sj. But if Sj changes any letter a 
 into b, while s^ changes b into c and Sg changes c into d, then S4 
 changes a into c and s^ changes b into d. Hence S4S3 and s^s^ 
 both change a into d ; and therefore, a being any letter operated 
 upon by the substitutions, 
 
 Hence the meaning of the symbol s^s^Ss is definite ; it 
 depends only on the component substitutions Sj, s^, S3 and their 
 sequence, and it is independent of the way in which they are 
 associated when their sequence is assigned. And the same 
 clearly holds for the symbol representing the successive per- 
 formance of any number of substitutions. To avoid circum- 
 locution, it is convenient to speak of the substitution SiS^-.-Sn 
 as the product of the substitutions Sj, s^,..., s„ in the sequence 
 given. The product of a number of substitutions, thus defined, 
 always obeys the a ssociati ve law but does not in general obey 
 the commutative law of algebraical multiplication. 
 
 7. The substitution which replaces every symbol by itself 
 is called the identical substitution. The inverse of a given 
 substitution is that substitution which, when performed after 
 the given substitution, gives as result the identical substitution. 
 Let s_i be the substitution inverse to s, so that, if 
 
 I a^ , ttg , . . > , ctjj 
 
 V61, 6a,..., bn. 
 
 then fh.,h,....K 
 
 \ai, aj,..., a, 
 
8] OF SUBSTITUTIONS 5 
 
 Let So denote the identical substitution which can be represented 
 
 by 
 
 /«!, a,,,..., an\ 
 \ai,a„,...,aj' 
 
 Then ss^^ — s^ and S-^s = So, '^ 
 
 so that s is the substitution inverse to s_]. 
 
 Now if ts = t's, , 
 
 theu tss-i = t'ss-T,, 
 
 or tSo = t'so. 
 
 But tso is the same substitution as t, since «„ prod uces no 
 change ; and therefore 
 
 t = t'. 
 
 In exactly the same way, it may be shewn that the relation 
 
 st = st' 
 
 involves t = t'. 
 
 8. The result of performing r times in succession the same 
 substitution s is represented symbolically by s''. Since, as 
 has been seen, products of substitutions obey the associative 
 law of multiplication, it follows that 
 
 g,J.gV = gH+V = gVg^_ 
 
 Now since there are only a finite number of distinct 
 
 substitutions that can be performed on a given finite set of 
 
 symbols, the series of substitutions s, s'\ s^,... cannot be all 
 
 distinct. Suppose that 5"*+^ is the first of the series which is 
 
 the same as s, so that 
 
 gm+i ^ g_ 
 
 Then s™ss-i = ss_i , 
 
 or s'" = Si). 
 
 There is no index fi smaller than m for which this relation 
 holds. For if 
 
 S'' = S'o, 
 
 then s'^+^ = sso = s, 
 
 contrary to the supposition that s™"'"^ is the first of the series 
 which is the same as s. 
 
6 NEGATIVE POWERS OF SUBSTITUTIONS [8 
 
 Moreover the m — 1 substitutions s, s',..., s*""' must be all 
 
 distinct. For if 
 
 s*" = s", v<fi<m, 
 
 then s'^-'s" (s")-! = s' («")_, , 
 
 or s''-'' = s„, 
 
 which has just been shewn to be impossible. 
 
 The number m is called the order of the substitution s. 
 In connection with the order of a substitution, two properties 
 are to be noted. First, if 
 
 it may be shewn at once that n is a multiple of m the order of 
 s ; and secondly, if 
 
 s" = s^, 
 
 then a -13 = (mod. m). 
 
 If now the equation 
 
 be assumed to hold, when either or both of the integers /j, and v 
 is a negative integer, a definite meaning is obtained for the 
 symbol s~^, implying the negative power of a substitution ; and 
 a definite meaning is also obtained for s". For 
 
 so that s-'' = (s'')_i. 
 
 Similarly it can be shewn that 
 
 «" = «„. 
 
 Since every power of s„ is the same as s^, and since 
 wherever «„ occurs in the symbol SiS^-.-Sn of a compound 
 substitution it may be omitted without affecting the result, it 
 is clear that no ambiguity will result from replacing So every- 
 where by 1 ; in other words, we may use 1 to represent the 
 identical substitution which leaves every letter unchanged. 
 But when this is done, it must of course be remembered that 
 the equation 
 
 is not a reducible algebraical equation, which is capable of being 
 written in the form 
 
 (« - 1) (s™-' + s™-'' + . . . + 1) = 0. 
 
10] SIMILAR SUBSTITUTIONS 7 
 
 Indeed the symbol s + s', where s and s are any two substi- 
 tutions, has no meaning. 
 
 9. If the cycles of a substitution 
 
 s = (pqr...s)(p'q...s')(p"q"...s")... 
 
 sontain m, m, m",... letters respectively, and if 
 
 s'' = l, 
 
 fi must be a common multiple of m, m', m",.... For s** changes 
 p into a letter /i places from it in the cyclical set p, q, r,..., s; 
 and therefore, if it changes p into itself, /j, must be a multiple 
 of m. In the same way, it must be a multiple of m, m",.... 
 Hence the order of s is the least common multiple of m, m', 
 m",.... 
 
 In particular, when a substitution consists of a single 
 cycle, its order is equal to the number of letters which it inter- 
 changes. Such a substitution is called a circular substitution. 
 
 A substitution, all of whose cycles contain the same 
 number of letters, is said to be regular in the letters which it 
 interchanges ; the order of such a substitution is clearly equal 
 to the number of letters in one of its cycles. 
 
 10. Two substitutions, which contain the same number of 
 cycles and the same number of letters in corresponding cycles, 
 are called similar. If s, s' are similar substitutions, so also 
 clearly are s'', s'*"; and the orders of s and s' are the same. 
 
 Let now 
 
 s = (cfpttg. . .a«) (ap'aq'...as'). . . 
 
 V6i, 63,..., hj 
 be any two substitutions. Then 
 
 Veil, a^,..., aj^^ \0i, 03,..., On/ 
 
 = {bpbg...bs)(bpbg'...h^)..., 
 the latter form of the substitution being obtained by actually 
 carrying out the component substitutions of the earlier form. 
 Hence s and t-^st are similar substitutions. 
 
8 PERMUTABLE SUBSTITUTIONS [10 
 
 Since s^s^ = sf^SiS^Si, 
 
 it follows that SiS^ and SjSi are similar substitutions and therefore 
 that they are of the same order. Similarly it may be shewn 
 that SiSiSs-.-Sn, s^Ss.-.SnSi,..., s„Si...S2*3 are all similar substi- 
 tutions. 
 
 It may happen in particular cases that s and t~^st are the 
 same substitution. When this is so, t and s are permutable, 
 that is, st and ts are equivalent to one another ; for if 
 
 s = t-^st, 
 then ts = St. 
 
 This will certainly be the case when none of the symbols 
 that are interchanged by t are altered by s ; but it may happen 
 when s and t operate on the same symbols. Thus if 
 
 s = (ab) (cd), t = (ac) (bd), 
 then st = (ad) (be) = ts. 
 
 Ex. 1. Shew that every regular substitution is some power of a 
 circular substitution. 
 
 Ex. 2. If s, s' are permutable regular substitutions of the same 
 Tnn letters of orders m and n, these numbers being relatively prime, 
 shew that ss' is a circular substitution in the mn letters. 
 
 Ex.3* If s =(123) (456) (789), 
 
 si = (147) (258) (369), , „-/ ; 
 
 (456) (798), SrJ/V 
 
 shew that s is permutable with both Sj and s^, and that it can be 
 formed by a combination of Sj and s^. 
 
 Ex. 4. Shew that the only substitutions of n given letters 
 which are permutable with a circular substitution of the n letters 
 are the powers of the circular substitution. 
 
 Ex. 5. Determine all the substitutions of the ten symbols 
 involved in 
 
 s = (abcde) (a^ySe) 
 which are permutable with s. 
 
 The determination of all the substitutions which are 
 permutable with a given substitution will form the subject 
 of investigation in Chapter X. 
 
 * It is often convenient to use digits rather tban letters for the purpose of 
 illustration. 
 
11] TKANSPOSITIONS 9 
 
 11. A circular substitution of order two is called a trans- 
 position. It may be easily verified that 
 
 {pqr...s) = {pq)(pr)...{ps), 
 
 30 that every circular substitution can be represented as a 
 product of transpositions ; and thence, since every substitution 
 is the product of a number of circular substitutions, every 
 substitution can be represented as a product of transpositions. 
 It must be remembered, however, that, in general, when a 
 substitution is represented in this way, some of the letters will 
 occur more than once in the symbol, so that the order in which 
 the constituent transpositions occur is essential. There is thus 
 a fundamental difference from the case when the symbol of a 
 substitution is the product of circulai- substitutions, no two of 
 which contain a common letter. 
 
 Since {p'^) = {pp') (pq) (pp'), 
 
 every transposition, and therefore every substitution of n 
 letters, can be expressed in terms of the n — 1 transpositions 
 
 (aiffla), (01^3),..., (a^an). 
 
 The number of different ways in which a given substitution 
 may be represented as a product of transpositions is evidently 
 unlimited; but it may be shewn that, however the representation 
 is effected, the number of transpositions is either always even 
 or always odd. To prove this, it is sufficient to consider the 
 effect of a transposition on the square root of the discriminant 
 of the n letters, which may be written 
 
 r = n—l ( s — n 
 
 D= li n {a.r-a,) 
 
 r=l [s=-r+l 
 
 The transposition (a^ag) changes the sign of the factor 
 ttr — ag. When q is less than either r or s, the transposition 
 interchanges the factors a^ - a,, and ctj - flg ; and when q is 
 greater than either r or s, it interchanges the factors a,- — Ug and 
 Us-ag. When q lies between r and s, the pair of factors 
 and aq — cvs are interchanged and are both changed in 
 
 ar — a, 
 
 g ClUU LI/j 
 
 sign. Hence the effect of the single transposition on B is 
 to change its sign. Since any substitution can be expressed as 
 the product of a number of transpositions, the effect of any 
 substitution on D must be either to leave it unaltered or to 
 
10 EVEN AND ODD SUBSTITUTIONS [11 
 
 change its sign. If a substitution leaves D unaltered it must, 
 when expressed as a product of transpositions in any way, 
 contain an even number of transpositions ; and if it changes 
 the sign of D, every representation of it, as a product of 
 transpositions, must contain an odd number of transpositions. 
 Hence no substitution is capable of being expressed both by an 
 even and by an odd number of transpositions. 
 
 A substitution is spoken of as odd or even, according as the 
 transpositions which enter into its representation are odd or 
 even in number. 
 
 Further, an even substitution can always be represented as 
 a product of circular substitutions of order three. For any. 
 even substitution of n letters can be represented as the product 
 of an even number of the n — \ transpositions 
 
 (cfiCts), (aia,),..., (OiCin), 
 in appropriate sequence and with the proper number of 
 occurrences ; and the product of any consecutive pair of these 
 («!«,.) {a^ag) is the circular substitution {a^aras). 
 
 Now (ai aa ««) («i aj o-r) {(ha^ cisY 
 
 = (aiajds) {aia^ar) {oiasa^) 
 = {(hara,), 
 so that every circular substitution of order three displacing aj, 
 and therefore every even substitution of n letters, can be 
 expressed in terms of the n — 2 substitutions 
 (oiaatts), (aia,ai),..., {(ha^ari) 
 and their powers. 
 
 Ex. 1. Shew that every even substitution of n letters can be 
 expressed in terms of 
 
 (oiajcts), {a^a^a^), , (aia„_ia„), 
 
 when n is odd ; and in terms of 
 
 {a^a^a^}, {a^aia^, , (ai«„_2a„-i), {a^a^a^), 
 
 when n is even. 
 
 -.^^> / 
 
 Ex. 2. If w + 1 is odd.^^shew that every even substitution of 
 mn + 1 letters can be expressed in terms of 
 
 (»i"2 '*K+i)' (%«)n-2 »2K+i)j , (»ia{m_i)„+2 am«+i); 
 
 and if »i + 1 is even, that every substitution of mn + 1 letters can 
 be expressed in terms of this set of in circular substitutions. 
 
CHAPTEK II. 
 
 THE DEFINITION OF A GROUP. 
 
 12. In the present chapter we shall enter on our main 
 subject and we shall begin with definitions, explanations and 
 examples of what is meant by a group. 
 
 Definition. Let 
 
 A,B,C,... 
 
 represent a set of operations, which can be performed on the 
 same object or set of objects. Suppose this set of operations 
 has the following characteristics. 
 
 (a) The operations of the set are all distinct, so that no 
 two of them produce the same change in every possible appli- 
 cation. 
 
 (;S) The result of performing successively any number of 
 operations of the set, say A, B,..., K, is a no^faw definite 
 operation of the set, which depends only on the component 
 operations and the sequence in which they are carried out, and 
 not on the way in which they may be regarded as associated. 
 Thus A followed by B and B followed by C are operations of 
 the set, say D and E ; and D followed by G is the same opera- 
 tion as A followed by E. 
 
 (7) A being any operation of the set, there is always 
 antAir operation A-i belonging to the set, such that A 
 followed by 4_i produces no change in any object. 
 
12 DEFINITION OF A GROUP [12 
 
 The operation A_i is called the inverse of A. 
 
 The set of operations is then said to form a Group. 
 
 From the definition of the inverse of A given in (7), it 
 follows directly that A is the inverse of J._i. For if A changes 
 any object O into O', J._i must change O' into O. Hence A_i 
 followed by A leaves il', and therefore every object, unchanged. 
 
 The operation resulting from the successive performance of 
 the operations A, B,..., K in the sequence given is denoted by 
 the symbol AB...K; and if O is any object on which the 
 operations may be performed, the result of carrying out this 
 compound operation on O is denoted by O . AB...K. 
 
 If the component operations are all the same, say A, and r 
 in number, the abbreviation A'' will be used for the resultant 
 operation, and it will be called the rth power oi A. 
 
 Definition. Two operations, A and B, are said to be 
 permutable when AB and BA are the same operation. 
 
 13. If AB and AC are the same operation, so also are 
 A^iAB and A-iAC. But the operation A^^A produces no 
 change in any object and therefore A_iAB and B, producing 
 the same change in every object, are the same operation. Hence 
 B and G are the same operation. 
 
 This is expressed symbolically by saying that, if 
 
 AB = AG, 
 
 then B = G; 
 
 the sign of equality being used to imply that the symbols 
 represent the same operation. 
 
 In a similar way, if 
 
 BA = GA, 
 it follows that B=G. 
 
 From conditions (/S) and (7), A A., must be a definite 
 operation of the group. This operation, by definition, pro- 
 duces no change in any possible object, and it must, by 
 condition (a), be unique. It is called the identical operation. 
 
14] CONTINUOUS AND DISCONTINUOUS GROUPS 13 
 
 If it is represented by A^ and if A be any other operation, 
 then 
 
 A„A = A = AA„, 
 
 and for every integer r, 
 
 Hence A^ may, without ambiguity, be replaced by 1, 
 wherever it occurs. 
 
 14. The number of distinct operations contained in a 
 group may be either finite or infinite. When the number is 
 infinite, the group may contain operations which produce an 
 infinitesimal change in every possible object or operand. 
 
 Thus the totality of distinct displacements of a rigid body 
 evidently forms a group, for they satisfy conditions (a), (0) and 
 (7) of the definition. Moreover this group contains operations 
 of the kind in question, namely infinitesimal twists ; and 
 each operation of the group can be constructed by the con- 
 tinual repetition of a suitably chosen infinitesimal twist. 
 
 Next, the set of translations, that arise by shifting a cube 
 parallel to its edges through distances which are any multiples 
 of an edge, forms a group containing an infinite number 
 of operations ; but this group contains no operation which 1 
 effects an infinitesimal change in the position of the cube. 
 
 As a third example, consider the set of displacements by 
 which a complete right circular cone is brought to coincidence 
 with itself. It consists of rotations through any angle about 
 the axis of the cone, and rotations through two right angles 
 about any line through the vertex at right angles to the 
 axis. Once again this set of displacements satisfies the con- 
 ditions (a), (/3) and (7) of the definition and forms a group. 
 
 This last group contains infinitesimal operations, namely 
 rotations round the axis through an infinitesimal angle; and 
 every finite rotation round the axis can be formed by the 
 continued repetition of an infinitesimal rotation. There is 
 however in this case no infinitesimal displacement of the group 
 by whose continued repetition a rotation through two right 
 angles about a line through the vertex at right angles to the 
 axis can be constructed. Of these three groups with an infinite 
 
14 ORDER OF A GROUP [14 
 
 number of operations, the first is said to be a contimious group, 
 the second a discontinuous group, and the third a mixed group. 
 
 Continuous groups and mixed groups lie entirely outside the 
 plan of the present treatise ; and though, later on, some of the 
 properties of discontinuous groups with an infinite number of 
 operations will be considered, such groups will be approached 
 from a point of view suggested by the treatment of groups 
 containing a finite number of operations. It is not therefore 
 necessary here to deal in detail with the classification of infinite 
 groups which is indicated by the three examples given above ; 
 and we pass on at once to the case of groups which contain a 
 finite number only of distinct operations. 
 
 15. Definition. If the number of distinct operations con- 
 tained in a group be finite, the number is called the order of the 
 group. 
 
 Let S be an operation of a group of finite order N. Then 
 the infinite series of operations 
 
 S, S\ S' 
 
 must all be contained in the group, and therefore a finite 
 
 number of them only can be distinct. If (S™^* is the first of 
 
 the series which is the same as 8, and if *S_i is the operation 
 inverse to S, then 
 
 or /S™'=1. 
 
 Exactly as in § 8, it may be shewn that, if 
 
 /S''=l, 
 
 fi must be a multiple of m, and that the operations S, S^ , 
 
 /Sf"^i are all distinct. 
 
 Since the group contains only N distinct operations, m 
 must be equal to or less than N. It will be seen later that, if 
 m is less than N, it must be a factor of N. 
 
 The integer m is called the order of the operation S. The 
 order m' of the operation S" is the least integer for which 
 
 that is, for which xm' = (mod. m). 
 
16] NEGATIVE POWERS OF AN OPERATION 15 
 
 Hence, if g is the greatest common factor of x and m, 
 
 , m 
 m = — ; 
 
 9 
 and, if m is prime, all the powers of ;S', whose indices are less 
 than m, are of order m. 
 
 Since /S*<Sf^»' = ,S™= 1, {x<m), 
 
 and S''{8'')_^ = \, 
 
 it follows that (/Sf*)_i = /S"^»^. 
 
 If now a meaning be attached to S~^, by assuming that 
 the equation 
 
 S'+y = B'Sy 
 
 holds when either a; or y is a negative integer, then 
 
 and (5[«)_, = ,§-»', 
 
 so that 8~" denotes the inverse of the operation (S* 
 
 Ex. If Sa, Si, , S^, S^ are operations of a group, shew that 
 
 the operation inverse to Sa'^S/ S.'^Sj is Sa-%-^ --V^'^^"'. 
 
 16. If 1, Si, Si, , Sn-1 
 
 are the N operations of a group of order N, the set of iV 
 operations 
 
 Sf, SrSi, Oy02, , Oy»Sijvr_i 
 
 are (§ 13) all distinct ; and their number is equal to the order of 
 the group. Hence every operation of the group occurs once 
 and only once in this set. 
 
 Similarly every operation of the group occurs once and only 
 once in the set 
 
 Or, OiOf, O^Sr, , Ojy_]0^. 
 
 Every operation of the group can therefore be represented 
 as the product of two operations of the group, and either the 
 first factor or the second factor can be chosen at will. 
 
 A relation of the form 
 
 Sp = Sq 8r 
 
 between three operations of the group will not in general 
 involve any necessary relation between the order of Sp and the 
 
16 PRODUCT OF PERMUTABLE OPERATIONS [16 
 
 orders of Sg and S,.. If however the two latter are permutable, 
 the relation requires that, for all values of ;», 
 
 and in that case the order of 8^ is the least common multiple 
 of the orders of /Sg and *S,.. r^ ?> e -^ -/ '•>- \ ' -,. , • *" 
 
 Suppose now that S, an operation of the group, is of -brderi 
 nin, where m and n are relatively prime. Then we may shew 
 that, of the various ways in which S may be represented as the 
 product of two operations of the group, there is just one in 
 which the operations are permutable and of orders m and.w 
 respectively. 
 
 Thus let /S™ = M, 
 
 and (Sf™ = N, 
 
 so that M, N are operations of orders m and n. Since iS™ and 
 /S" are permutable, so also are M and N, and powers of M and N. 
 
 If x„, yo are integers satisfying the equation 
 xn + ym = 1, 
 every other integral solution is given by 
 
 x = Xa + tm, y = ya~tn, 
 where t is an integer. 
 
 Now M'^Ny = S'^'^+y"' = S ; 
 
 and since x and m are relatively prime, as also are y and m, 
 M^ and N^ are permutable operations of orders m and n, so 
 that S is expressed in the desired form. 
 
 Moreover, it is the only expression of this form ; for let 
 
 where M^ and Ni are permutable and of orders m and ii. 
 
 Then /S" = il/i», since N^'^ = l. 
 
 Hence ilfi" = M, 
 
 or Ml*" = M*, 
 
 or Mi^-v^=M='. 
 
 But JIfi™ = 1, and therefore ifr"'" = 1 ; hence 
 
 M, = JW*. 
 In the same way it is shewn that N^ is the same as N^. The 
 representation of *Si in the desired form is therefore unique. 
 
17] EXAMPLES OF GROUPS 17 
 
 17. Two given operations of a group successively performed 
 give rise to a third operation of the group which, when the 
 operations are of known concrete form, may be determined by 
 actually carrying out the two given operations. ThTis the set 
 of finite rotations, which bring a regular solid to coincidence 
 with itself, evidently form a group ; and it is a purely geo- 
 metrical problem to determine that particular rotation of the 
 group which arises from the successive performance of two 
 given rotations of the group. 
 
 When the operations are represented by symbols, the rela- 
 tion in question is represented by an equation of the form 
 
 AB=C; 
 but the equation indicates nothing of the nature of the actual 
 operations. Now it may happen, when the operations of two 
 groups of equal order are represented by symbols, 
 
 (i) 1, A, B, C 
 
 (ii) 1, A', B', C, 
 
 that, to every relation of the form 
 
 AB^G 
 between operations of the first group, there corresponds the 
 relation 
 
 A'F = C 
 
 between operations of the second group. In such a case, 
 although the nature of the actual operations in the first group 
 may be entirely different from the nature of those in the 
 second, the laws according to which the operations of each 
 group combine among themselves are identical. The following 
 series of groups of operations, of order six, will at once illustrate 
 the possibility just mentioned, and will serve as concrete exam- 
 ples to familiarize the reader with the conception of a group of 
 operations. 
 
 I. Group of inversions. Let P, Q, R be three circles with 
 a common radical axis and let each pair of them intersect at an 
 angle ^tt. Denote the operations of inversion with respect to 
 P, Q, Rhy G,D, E; and denote successive inversions at P, R 
 and at P,Qhy A and B. The object of operation may be any 
 point in the plane of the circles, except the two common points 
 P. 2 
 
18 EXAMPLES [1' 
 
 in which they intersect. Then it is easy to verify, from the 
 geometrical properties of inversion, that the operations 
 
 1, A, 5, C, D, E 
 are all distinct, and that they form a group. For instance, DE 
 represents successive inversions at Q and R. But successive 
 inversions at Q and R produce the same displacement of points 
 as successive inversions at P and Q, and therefore 
 
 DE=B. 
 
 II. Group of rotations. Let POP', QOQ', ROR' be three 
 concurrent lines in a plane such that each of the angles POQ 
 and QOR is Att, and let lOI' be a perpendicular to their plane. 
 Denote by J. a rotation round //' through |7r bringing PP' to 
 RR'; and by £ a rotation round //' through ^ir bringing PP 
 to Q'Q. Denote also by C, D, E rotations through two right 
 angles round PP', QQ', RR'. The object of the rotations may 
 be any point or set of points in space. Then it may again be 
 verified, by simple geometrical considerations, that the opera- 
 tions 
 
 1, A, B. G, D, E 
 
 are distinct and that they form a group. 
 
 III. Group of linear transformations of a single variable. 
 The operation of replacing !c by a given function /(«) of itself 
 is sometimes represented by the symbol (x, fix)). With this 
 notation, if 
 
 E={x,-^^~^, l={x,x), 
 
 it may again be verified without difficulty that these six 
 operations form a group. 
 
 IV. Group of linear transformations of two variables. 
 With a similar notation, the six operations 
 
 form a group. 
 
17] OF GROUPS 19 
 
 V. Group of linear transformations to a prime modulus. 
 The six operations defined by 
 
 A={x,x+l), B = {x,x + 2), C = {x,^x), 
 
 D = (x,2x+'i), E = {x,2x^-l), l={x,x), 
 
 where each transformation is taken to modulus 8, form a group. 
 
 VI. Group of substitutions of ^ symbols. The six substitu- 
 tions 
 
 1, A={xyz), B = {xzy), C = x(yz), D = y(zx), E = z(xy) 
 
 are the only substitutions that can be formed with three symbols; 
 they must therefore form a group. 
 
 VII. Groiip of substitutions of 6 symbols. The substitu- 
 tions 
 
 1, A= (xyz) (abc), B — (xzy) (acb), C = (xa) (yc) (zb), 
 
 D = xb (ya) (zc), E = (xc) (yb) (za) 
 
 may be verified to form a group. 
 
 VIII. Group of substitutions of 6 symbols. The substitu- 
 tions 
 
 1, A = (xaybzc), B = (xyz) (abc), C = (xb) (yc) (za), 
 
 D = (xzy) (acb), E = (xczbya) 
 form a group. 
 
 The operations in the first seven of these groups, as well as 
 the objects of operation, are quite different from one group to 
 another ; but it may be shewn that the laws according to which 
 the operations, denoted by the same letters in the different 
 groups, combine together are identical for all seven. There is 
 no difficulty in verifying that in each instance 
 
 ^3 = 1, 0^=1, B = A\ D = AC=OA\ E = A'G=CA; 
 
 and from these relations the complete system, according to 
 which the six operations in each of the seven groups combine 
 together, may be at once constructed. This is given by the 
 following multiplication table, where the left-hand vertical 
 column gives the first factor and the top horizontal line the 
 
 2—2 
 
20 
 
 MULTTPI.ICATION TABLE OF A GROUP 
 
 [17 
 
 second factor in each product; thus the table is to be read 
 Al = A, AB = 1, AC= D, and so on. 
 
 BODE 
 
 1 
 A 
 B 
 
 a 
 
 D 
 
 E 
 
 \ A B C D E 
 
 A B I D E C 
 
 B \ A E C D 
 
 C E D 1 B A 
 
 D C E A IB 
 
 E B C B A 1 
 
 But, though the operations of the seventh and eighth 
 groups are of the same nature and though the operands are 
 identical, the, laws according to which the six operations 
 combine together are quite distinct for the two groups. Thus, 
 for the last group, it may be shewn that 
 
 B = A\ C = A\ B = A\ E = A\ ^" = 1, 
 so that the operations of this group may, in fact, be represented 
 
 by 
 
 1, A, A\ A\ A\ A\ 
 
 18. If we pay no attention to the nature of the actual 
 operations and operands, and consider only the number of the 
 former and the laws according to which they combine, the first 
 seven groups of the preceding paragraph are identical with 
 each other. From this point of view a group, abstractly 
 considered, is completely defined by its multiplication table; 
 and, conversely, the multiplication table must implicitly 
 contain all properties of the group which are independent of 
 any special mode of representation. 
 
 It is of course obvious that this table cannot be arbitrarily 
 constructed. Thus, if 
 
 AB = P and BC=Q, 
 the entry in the table for PC must be the same as that for AQ. 
 Except in the very simplest cases, the attempt to form a 
 consistent multiplication table, merely by trial, would be most 
 laborious. 
 
19] DEFINING RELATIONS OF A GROUP 21 
 
 The very existence of the table shews that the symbols 
 denoting the different operations of the group are not all 
 independent of each other ; and since the number of symbols is 
 finite, it follows that there must exist a set of symbols 
 Si, S^,..., Sn no one of which can be expressed in terms of the 
 remainder, while every operation of the group is expressible in 
 terms of the set. Such a set is called a set of fundamental or 
 generating operations of the group. Moreover though no one 
 of the generating operations can be expressed in terms of the 
 remainder, there must be relations of the general form 
 
 among them, as otherwise the group would be of infinite order ; 
 and the number of these relations, which are independent 
 of one another, must be finite. Among them there necessarily 
 occur the relations 
 
 ,S/. = 1, S/^ = l, , S™«« = 1, 
 
 giving the orders of the fundamental operations. 
 
 We thus arrive at a virtually new conception of a group ; 
 it can be regarded as arising from a finite number of funda- 
 mental operations connected by a finite number of independent 
 relations. But it is to be noted that there is no reason for 
 supposing that such an origin for a group is unique ; indeed, 
 in general, it is not so. Thus there is no difficulty in verifying 
 that the group, whose multiplication table is given in § 17, is 
 completely specified either by the system of relations 
 
 ^3=1^ 0^ = 1, {ACy=l, 
 or by the system 
 
 In other words, it may be generated by two operations of 
 orders 2 and 3, or by two operations of order 2. So also the 
 last group of § 17 is specified either by 
 
 A'=l, 
 or by 5'=1, C^ = l, BG=GB. 
 
 19. Definition. Let G and 0' be two groups of equal 
 order. If a correspondence can be established between the 
 operations of G and G', so that to every operation of G there 
 
22 
 
 SiMfLY ISOMORPHIC GROUPS [19 
 
 corresponds a single operation of G' and to every operation of 
 G' there corresponds a single operation of G, while to the 
 product AB of any two operations of G there corresponds the 
 product A'B' of the two corresponding operations of G', the 
 groups G and G' are said to be simply isomorphic^. 
 
 Two simply isomorphic groups are, abstractly considered, 
 identical. In discussing the properties of groups, some definite 
 mode of representation is, in general, indispensable; and as 
 long as we are dealing with the properties of a group per se, 
 and not with properties which depend on the form of 
 representation, the group may, if convenient, be replaced by 
 any group which is simply isomorphic with it. For the dis- 
 cussion of such properties, it would be most natural to suppose 
 the group given either by its multiplication table or by its 
 fundamental operations and the relations connecting them; 
 and as far as possible we shall follow this course. Unfortu- 
 nately, however, these purely abstract modes of representing a 
 group are by no means the easiest to deal with. It thus 
 becomes an important question to determine as far as possible 
 what different concrete forms of representation any particular 
 group may be capable of; and we shall accordingly end the 
 present chapter with a demonstration of the following general 
 theorem bearing on this question. 
 
 20. Theorem. Every group of finite order N is capable of 
 representation as a group of substitutions of N symbols^ 
 
 Let 1, 8i, fSa,..., Si,..., S^f-i 
 
 be the N operations of the group ; and form the complete 
 multiplication table 
 
 I , 8, , 82 ,..., 8i ,..., (Sjf_i , 
 81 , Si' , S2S1 ,..., SiSi ,..., S^-iSi, 
 
 Si , SiSi , S^Si ,..., Si" ,..., Sjif^iSi , 
 
 8n-i, SiS^^i, (SaOjy-],..., 8iSjf_i,... , oV-i > 
 
 1 We shall sometimes use the phrase that two groups are of the same type 
 to denote that they are simply isomorphic. 
 
 = Dyck, "-Gruppentheoretisehe Studien," Math. Ann., Vol. xx (1882), p. 30. 
 
20] dyck's theorem 23 
 
 that results from multiplying the s^'mbols in the first horizontal 
 line by 1, 8i, 8^,..., /Sjf_i in order. Each horizontal line in the 
 table so obtained contains the same N symbols as the original 
 line, but any given symbol occupies a different place in each 
 line ; for the supposition that the two symbols SiSp and SiSq 
 were identical would involve 
 
 which is not true. 
 
 It thus appears that the first line in the table, taken 
 with the (i + l)th line, defines a substitution 
 
 / 1 , Si , 82 ,..., S^f-i \ 
 \Si, Si8i, S^Si,..., Sjf-iSiJ 
 
 performed on the N symbols 1, Si, 8.,,..., S^-i', and by taking 
 the first line with each of the others, including itself, a set of N 
 substitutions performed on these iV symbols is obtained. Now 
 this set of substitutions forms a group simply isomorphic 
 to the given group. For if 
 
 SpSi = Sq and SqSj = S,- , 
 the substitutions 
 
 /I , Si , 8-2 ,..., Sjf-i \ 
 \8i, SiSi, S^Si,..., Sn—ioJ 
 
 /I, 81 , 82 ,..., Sjf-i \ 
 
 anCl \ c, rf r, cf CI CY O I ' 
 
 VOj, OiOj, O2OJ,..., I02f—i0j/ 
 
 successively performed, change 8p into /S,.. 
 
 But SpSiSj = S,-, 
 
 and therefore the product of the two substitutions, in the order 
 given, is the substitution 
 
 / 1 , 81 , »S'„ ,..., Sn-1 \ 
 \SiSj, SiSiSj, S-iSiSj,..., 8j<f-i8i8j/ 
 
 or the product of any two of the N substitutions is again one 
 of the substitutions of the set. Moreover the above reasoning 
 shews that the result of performing successively the substitu- 
 tions corresponding to the operations 8,; and Sj gives the 
 substitution corresponding to the operation SiSj. Hence, 
 since the number of substitutions is equal to the number of 
 
24 dyck's theohem [20 
 
 operations, the given group and the group of substitutions 
 are simply isomorphic. 
 
 It may also be shewn that each of the N substitutions is 
 regular (§ 9) in the N symbols. For the substitution 
 
 \Si, SiSi, SiSi,..., S-fT-iSj 
 
 changes S,- into S^St, SrSi into SrSi', and so on. If then m is 
 the order of Si, the cycle of the substitution which contains S^ 
 will be 
 
 (Sr, SrSi, SrSe,..., SrSi"'-'). 
 
 Now 8r may be any symbol of the- set ; hence all the cycles 
 of the substitution must contain the same number, m, of symbols. 
 
 The substitution is therefore regular in the iV symbols, and 
 this can only be the case if ?w is a factor of N. It follows at 
 once, as was stated in § 15, that the order of any operation of a 
 group of order N must be equal to or a factor of K 
 
 All the substitutions in this form of representing a group 
 being regular, the group itself is said to be expressed as a 
 regular substitution group. 
 
 21. The form, in which it has just been shewn that every 
 group can be represented, is by no means the only form of 
 representation possessing this property. Thus Hurwitz' has 
 shewn that every group can be expressed as a group of rational 
 and reversible transformations which change a suitably chosen 
 algebraic curve (or Riemann's surface) into itself We shall see 
 in Chapter XIV that every group can also be expressed as a 
 group of linear transformations of a finite set of variables to a 
 prime modulus. 
 
 ^ " Algebraische Gebilde mit eindeutigen Transformationeu in sich," Math. 
 Ann., Vol. xli (1893), p. 421. 
 
CHAPTER III. 
 
 ON THE SIMPLER PROPERTIES OF A GROUP WHICH 
 ARE INDEPENDENT OF ITS MODE OF REPRESENTATION. 
 
 22. In this chapter we proceed to discuss some of the 
 simplest of the properties of groups of finite order which are 
 independent of their mode of representation. 
 
 If among the operations of a group a certain set can be 
 chosen which do not exhaust all the operations of the group G, 
 yet which at the same time satisfy all the conditions of § 12 so 
 that they form another group H, this group H is called a sub- 
 group of the group G. Thus if S be any operation, order m, 
 of 0, the operations 
 
 1, ^, S-\ , /S— 1 
 
 evidently form a group : and when the order of G is greater 
 than m, this group is a sub-group of G. A sub-group of this 
 nature, which consists of the different powers of a single 
 operation, is called a cyclical sub-group; and a group, which 
 consists of the different powers of a single operation, is called 
 a cyclical group. 
 
 Theorem I. If H is a sub-group of G, the order n of H is 
 a factor of the order N of G. 
 
 Let 1, Z, T,, , y„_, 
 
 be the n operations of H ; and let S^ be any operation of G 
 which is not contained in H. 
 
26 
 
 SUB-GHOUPS [22 
 
 Then the operations 
 
 8u TA, TA, , Tn^A 
 
 are all distinct from each other and from the operations of H. 
 
 For if TpSi = TqST,, 
 
 then Tp=Tq, 
 
 contrary to supposition ; and if 
 
 J-p ^ IqOi, 
 
 then 8i = Tq~^Tp, 
 
 and Si would be contained among the operations of H. 
 
 If the 2w operations thus obtained do not exhaust all the 
 operations of G, let S^ be any operation of G not contained 
 among them. 
 
 Then it may be shewn, by repeating the previous reasoning, 
 that the n operations 
 
 are all different from each other and from the previous 2« 
 operations. If the group G is still not exhausted, this process 
 may be repeated ; so that finally the N operations of G can be 
 exhibited in the form 
 
 1 T T T 
 
 Oi , JiOi , -ta'Ji , , -ij[_iOi 
 
 Si , T^S^ , T2S., , , Tf^iS^ 
 
 Hence N= nin, and n is therefore a factor of N. 
 
 When iV is a prime ^j, the group G can have no sub-group other 
 than one of order unity consisting of the identical operation alone. 
 Every operation S of the group, other than the identical operation, 
 is of order p, and the group consists of the operations 
 
 1, s, s% , sp-\ 
 
 A group whose order is prime is therefore necessarily cyclical. 
 
 23. Theorem II. The operations common to ttvo groups 
 Gi, G.. themselves form a group g, whose order is a factor' of the 
 orders of G^ and G... 
 
24] NOTATION FOR REPRESENTING GROUPS 27 
 
 For if ;Si, T are any two operations common to 0-^ and G^, 
 ST is also common to both groups ; and hence the common 
 operations satisfy conditions (a) and (/3) of the definition in § 12. 
 But their orders are finite and they must therefore satisfy also 
 condition (7), and form a group g. Moreover ^ is a sub-group 
 of both Gi and G^, and therefore by Theorem I its order is a 
 factor of the orders of both these groups. 
 
 If Gi and G^ are sub-groups of a third group G, then g is 
 also clearly a sub-group of G. 
 
 The set of operations, that arise by combining in every way 
 the operations of the groups Gi and G-i, evidently satisfy the 
 conditions of § 12 and form a group ; but this will not necessarily 
 or generally be a gi-oup of finite order. If however G^ and G2 
 are sub-groups of a group G of finite order, the group g' that 
 arises from their combination "will necessarily be of finite order; 
 it may either coincide with (? or be a sub-group of G. In 
 either case, the order of g' will be a multiple of the orders of 
 Gi and (?2- 
 
 It is convenient here to explain a notation that enables 
 us to avoid an otherwise rather cumbrous phraseology. Let 
 
 Si, 82, S3, be a given set of operations, and Gi, G.^, a 
 
 set of groups. Then the symbol 
 
 {Si, S2, S3, , Gi, G2, } 
 
 will be used to denote the group that arises by combining in 
 every possible way the given operations and the operations of 
 the given groups. 
 
 Thus, for instance, the group g' above would be represented 
 by {Gi, G2]; 
 
 the cyclical group that arises from the powers of an operation 
 .Sfby {S}; 
 
 and, as a further example, the sixth group of § 17 may be 
 represented by 
 
 {(ocy), (xz)}. 
 
 24. Definition. If S and T are any two operations of a 
 group, the operations S and T-^ST are called conjugate opera- 
 tions; while T~^ST is spoken of as the result of transforming 
 the operation »S' by T. 
 
28 CONJUGATE OPERATIONS [24 
 
 The two operations 8 and T^hST are identical only when S 
 and T are permutable. For if 
 
 8=T-^ST, 
 then TS = ST. 
 
 Two conjugate operations are always of the same order. 
 For 
 
 (T-^ST*)" = T-^ST . T-^ST f-'ST 
 
 Therefore, if *S«=1, 
 
 and conversely, if 
 
 (T-'ST)'' = 1, 
 then <S'» = T . T-'S''T. T'^ = T{T-'-STY T-^ = Tf-^ = 1. 
 
 The operations ST and TS are always conjugate and 
 therefore of the same order; for 
 
 ST=T-'T.ST=T-KTS.T. 
 
 Ex. Shew that the operations S^S^ S^_iSn a,nd SrS^^,^ 
 
 »S',j»Si S,._i are conjugate within the group {S^, S2, , S^]. 
 
 Definition. An operation >Sf of a group G, which is 
 identical with all its conjugate operations, is called a self- 
 coiijuffdte operation. Such an operation must evidently he 
 permutable with each of the operations of G. 
 
 In every group the identical operation is self -con jugate ; and in 
 a group, whose operations are all permutable, every operation is 
 self-conjugate. A simple example of a group, which contains self- 
 conjugate operations other than the identical operation, while at the 
 same time its operations are not all self -conjugate, is given by 
 
 1(1234), (13)}. 
 
 It is easy to shew that the order of this group is 8, and that (13) (24) 
 is a self-conjugate operation. 
 
 If all the operations of a group be transformed by a given 
 operation, the set of transformed operations form a group. For 
 if Ti and T^ are any two operations of the group, so that T/R is 
 also an operation of the group, then 
 
 ^'-■ T, S . ,s'-> T,S = ,S'-' T, r,s ; 
 
25] CONJUGATE SUB-GROUPR 20 
 
 hence the product of any two operations of the transformed 
 set is an|ii^ operation belonging to the transformed set, and 
 the set therefore forms a group. Moreover the preceding 
 equation shews that the new group is simply isomorphic 
 to the original group. If G is the given group, the symbol 
 8~^GS will be used for the new group. When 8 belongs to the 
 group G, the groups G and 8~'^ G8 are evidently the same. 
 
 Now unless 8 is a. self-conjugate operation of G, the pairs of 
 operations T and 8~'^T8 will not all be identical when for T the 
 different operations of G are put in succession. Hence the 
 process of transforming all the operations of a group by one of 
 themselves is equivalent to establishing a correspondence 
 between the operations of the group, which exhibits it as 
 simply isomorphic with itself. 
 
 Definitions. When if is a sub-group of G and 8 is any 
 operation of G, the groups H and 8~^HS are called conjugate 
 sub-groups of G. 
 
 If H and S~^HS are identical, 8 is said to be permutable 
 with the sub-group H. This does not necessarily involve that 
 8 is permutable with each of the operations of H. 
 
 If H and 8~^B'8 are identical, whatever operation 8 is of 
 0, H is said to be a self -conjugate sub-group of G. 
 
 A group is called composite or simple, according as it does 
 or does not possess at least one self-conjugate sub-group other 
 , than that formed of the identical operation alone. 
 
 25. Theorem III. The operations of a group G, which 
 are permutable with a given operation T, form a sub-group H ; 
 and the order of G divided by the order of H is the number of 
 operations conjugate to T*. 
 
 If Rj and R.^ are any two operations permutable with T, so 
 that R^T=TBa and R^T=TR^; 
 
 then BAT=R/fR, = TR,R„ 
 
 and therefore RiR^ is permutable with T. The operations 
 permutable with T therefore form a group H. Let n be its 
 
 order and 1, R^, R^, , iJ„_i 
 
 * Among these T of course occurs. 
 
30 OPERATIONS PERMUTABLE WITH [25 
 
 its operations. Then if S is any operation of G not contained 
 in H, the operations 
 
 S, RiS, R2S, , jB,i_io 
 
 all transform T into the same operation T'. 
 
 For (RiS)-' TRi S = S'^Rf-' TRiS = S-' TS. 
 
 Also the n operations thus obtained are the only opera- 
 tions which transform T into T' ; for if 
 
 then SS'-'TS'S-^ = ST'S'' = T ; 
 
 and therefore S'S~'^ belongs to H. The number of operations 
 
 which transform T into any operation conjugate to it is 
 
 therefore equal to the number that transform T into itself, that 
 
 is, to the order of H. If then iV is the order of G, the 
 
 N 
 operations of G may be divided into — sets of n each, such 
 
 that the operations of each set transform T into a distinct 
 
 operation, those of the first set, namely the operations of H, 
 
 transforming T into itself The number of operations conjugate 
 
 N 
 to T, including itself, is therefore — . 
 
 Since T=ST'S-\ 
 
 therefore Rr'TRi = Rr'ST'S-'Ri ; 
 
 hence T'= S-'T8 = S'^Ri-'TRiS = S-'Rr'S . T . S-'RiS, 
 
 so that every operation of the form S'^RiS is permutable with 
 T'. Hence if H is the group of operations permutable with T, 
 
 and if S-'TS=T', 
 
 then S~^ffS is the group of operations permutable with T'. 
 
 It is convenient to have a symbol to represent the set of 
 operations 
 
 >S^, RiS, R^S,..., -Kn_iO, 
 
 where 1, R, , R^,..., R,^, 
 
 form a group H. We shall in future represent this set of 
 operations by HS ; and we shall use SH to represent the set 
 
 S, SRi, SR.;,..., SRji_j. 
 
26] A GIVEN OPERATION OR SUB-GROUP 31 
 
 Theorem IV. The operations of a group G which are 
 permutable with a sub-group H form a sub-group I, which is 
 either identical with H or contains H as a self-conjugate sub- 
 group. The order of divided by the order of I is the number 
 of sub-groups conjugate to H* 
 
 If S-i, S2 are any two operations of G which are pernrn table 
 with H, then 
 
 and therefore Sr^-'HS^S, = H, 
 
 so that 8^82 is permutable with H. The operations of G 
 which are permutable with H therefore form a group I, which 
 may be identical with H and, if not identical with H, must 
 contain it. Also, if 8 is any operation of I, 
 
 8-'HS = H, 
 
 and therefore H is a. self-conjugate sub-group of I. 
 
 If now 2 is any operation of G not contained in /, it may 
 be shewn, exactly as in the proof of Theorem III, that the 
 operations /S and no others transform H into a conjugate 
 sub-group H' which is not identical with H; and therefore 
 that the number of sub-groups in the conjugate set to which 
 H belongs is the order of G divided by the order of /. 
 
 The operations of G which are permutable with H' may 
 also be shewn to form the group 2~^72. 
 
 It is perhaps not superfluous to point out that two 
 distinct conjugate sub-groups may have some operations in 
 common with one another. 
 
 26. Let 8t, be any operation of G, and 
 
 Si, 82, , Sm 
 
 the distinct operations obtained on transforming 81 by every 
 operation of G. The number, m, of these operations is, by 
 Theorem III, a factor of N, the order of G. Moreover if, 
 instead of transforming Si, we transform any other operation of 
 the set, 8r, by every operation of G, the same set of to distinct 
 operations of G will result. Such a set of operations we call a 
 complete set of conjugate operations. If T is any operation of 
 * Among these sub-groups H itself occurs, 
 
32 COMPLETE SET OF CONJUGATE OPERATIONS [26 
 
 G which does not belong to this complete set of conjugate 
 operations, no operation that is conjugate to T can belong to 
 the set. Hence the operations of G may be distributed into a 
 number of distinct sets such that every operation belongs to 
 one set and no operation belongs to more than one set ; while 
 any set forms by itself a complete set of conjugate operations. 
 
 If TOi,Wo, ,ms are the numbers of operations in the different 
 
 sets, then 
 
 iV = mj + TOa + + »is ; 
 
 and, since the identical operation is self-conjugate, one at least 
 of the m's. must be unity. 
 
 Similarly, if H^ is any sub-gi'oup of G, and 
 
 H-i, H2, , Hp 
 
 the distinct sub-groups obtained on transforming H^ by every 
 operation of G, we call the set a complete set of conjugate 
 sub-groups. If iT is a sub-group of G not contained in the set, 
 no sub-group conjugate to K can belong to the set. If the 
 operation S-^ belongs to one or more of a complete set of 
 conjugate sub-groups, S~'*Sfi2 must also belong to one or more 
 sub-groups of the set, S being any operation of G. Hence 
 among the operations contained in the complete set of con- 
 jugate sub-groups, the complete set of conjugate operations 
 
 occurs. 
 
 No sub-group of G can contain operations belonging to 
 every one of the complete sets of conjugate operations of G. 
 For if • such a sub-group H existed, the complete set of 
 conjugate sub-groups, to which H belongs, would contain all 
 the operations of G. Let m be the order of H and n (> m) the 
 order of the sub-group I formed of those operations of G which 
 
 N 
 are permutable with H. Then H is one of — conjugate sub- 
 groups, each of which contains m operations. The identical 
 operation is common to all these sub-groups, and they therefore 
 cannot contain more than 
 
 1 I Njm- l) 
 
27] SELF-CONJUGATE SUB-GROUPS 33 
 
 distinct operations in all. This number is less than N, and 
 therefore the complete set of conjugate sub-groups cannot 
 contain all the operations of 0. 
 
 27. If a group contains self-conjugate operations, it must 
 contain self- conjugate sub-groups. For the cyclical sub-group 
 generated by any self-conjugate operation must be self- 
 conjugate. The only exception to this statement is the case of 
 the cyclical groups of prime order. Every operation of such a 
 group is clearly self-conjugate ; but since the cyclical sub-group 
 generated by any operation coincides with the group itself, 
 there can be no self-conjugate sub-group^ 
 
 If every operation of a group is not self-conjugate, or, in 
 other words, if the operations of a group are not all permutable 
 with each other, the totality of the self-conjugate operations 
 forms a self-conjugate sub-group. For, if S^ and S^ are 
 permutable with every operation of the group, so also is S-iS^. 
 
 Theorem V. The operations common to a complete set of 
 conjugate sub-groups form a self-conjugate sub-group. 
 
 It is an immediate consequence of Theorem II that the 
 operations common to a complete set of conjugate sub-groups 
 form a sub-group. Also the set of conjugate sub-groups, when 
 transformed by any operation of the group, is changed into 
 itself Hence their common sub-group must be self-conjugate. 
 
 It may of course happen that the identical operation is the 
 only one which is common to every sub-group of the set. 
 
 Corollary. The operations permutable with each of a 
 complete set of conjugate sub-groups form a self-conjugate 
 sub-group. 
 
 For, if the operations permutable with the sub-group H 
 form a sub-group /, the operations permutable with every 
 sub-group of the conjugate set to which H belongs are the 
 operations common to every sub-group of the conjugate set to 
 which I belongs. 
 
 -"O" 
 
 ' strictly speaking, this statement should be qualified by the addition 
 "except that formed by the identical operation alone." No real ambiguity 
 however will be introduced by always leaving this exception unexpressed. 
 
 B. 3 
 
31 
 
 SELF-CONJUGATE SUB-GROUPS [27 
 
 Further, the operations which are permutable with every 
 operation of a complete set of conjugate sub-groups fonn a 
 self-conjugate sub-group. 
 
 Theorem VI. If Ti, T^,..., T,. are a complete set of conjugate 
 operations of G, the group {Ti, T^,..., T,.}, if it does not coincide 
 with G, is a self -conjugate sub-group of G ; and it is the self- 
 conjugate suh-group of smallest order that contains T^. 
 
 Since the operations ^i, T.^,..., T,. are merely rearranged 
 in a new sequence when the set is transformed by any operation 
 of G, it follows that 
 
 S-^T„T.„...,T,.}S={T„T„...,Tr}, - 
 
 whatever operation of G may be represented by S. Hence 
 {Ti, Ti,..., Tr} is a self-conjugate sub-group. Also any self- 
 conjugate sub-group of G that contains 2\ must contain 
 T2, T3,..., T,.; and therefore any self-conjugate sub-group of G 
 which contains Tj must contain {T^, T^,..., Tr]. 
 
 In exactly the same way it may be shewn that, if 
 
 axe a complete set of conjugate sub-groups of G, the group 
 {jEfi, H^,..., Hg\, if it does not coincide with G, is the smallest 
 self-conjugate sub-group of G which comtains the sub-group Hi. 
 
 The theorem just proved suggests a process for determining 
 whether any given group is simple or composite. To this end, 
 the groups [T^, T^,-.., Tr} corresponding to each set of conjugate 
 operations in the group are formed. If any one of them 
 differs from the group itself, it is a self-conjugate sub-group 
 and the group is composite ; but if each group so formed 
 coincides with the original group, the latter is simple. If 
 the order of T^ contains more than one prime factor and 
 if Ti'" is of prime order, it is easy to see that the distinct 
 operations of the set Ti"^, T^,..., Ty™- form a complete set of 
 conjugate operations, and that the group \Ti,T^,..., Tr\ contains 
 the group {T™, T^,..., Tr"^]. Hence practically it is sufficient 
 to form the groups [T-^, T.^,..., T,] for all conjugate sets of opera- 
 tions whose orders are prime. 
 
 With the notation of § 26 (p. 32), the order of any self- 
 conjugate sub-group of G must be of the form m.+mB + ; 
 
28] MAXIMUM SUB-GROUPS 3o 
 
 for if the sub-group contains any given operation, it must contain all 
 the operations conjugate with it. Moreover one at least of the 
 
 numbers ot», 7n^, must be unity, since the sub-group must 
 
 contain the identical operation. It may happen that the numbers 
 
 m^ are such that the only factors of JS^ of the form m„, + m^+ , 
 
 one of the m's being unity, are JV itself and unity. When this is 
 the case, G is necessarily a simple group. It must not however 
 be inferred that, if iV has factors of this form, other than JV itself 
 and unity, then G is necessarily composite. 
 
 If Gi and 0-2 are sub-groups of G, it has already been seen 
 (§ 23) that the operations common to G^ and G.2 form a sub- 
 group g oi G ; and it is now obvious that, when G^ and G« are 
 self-conjugate sub-groups, so also is g. Moreover the group 
 {Gi, G^} is a self-conjugate sub-group unless it coincides with 
 G. For 
 
 8-' [G„ G.2\ 8= {8-'G,8, 8-'G,8} = \G„ G,}. 
 
 Again, with the same notation, if T^ is an operation of G 
 not contained in the self-conjugate sub-group G^, and if 
 Ti, T „,..., T,- is a complete set of conjugate operations, the 
 group {(?i, Ti, T^,..., T,.} is a self-conjugate sub-group, unless 
 it coincides with G. 
 
 Definitions. If (?j, a self-conjugate sub-group of G, is 
 such that the group 
 
 coincides with G, when Ti, T^,..., T^ is any complete set of 
 conjugate operations not contained in G-^, then G^ is said to be 
 a maodmum self-conjugate sub-group of G. This does not 
 imply that G^ is the self-conjugate sub-group of G of absol- 
 utely greatest order; but that there is no self-conjugate sub- 
 group of G;, distinct from G itself, which contains Gi and is of 
 greater order than Gi . 
 
 If fl^ is a sub-group of G, and if, for every operation 8 of G 
 which does not belong to II, the group {H, 8] coincides with G, 
 H is said to be a maodmum sub-group of G. 
 
 28. Definition. When a correspondence can be estab- 
 lished between the operations of a group G and the operations 
 of a group G', whose order is smaller than the order of G, such 
 that to each operation 8 of G there corresponds a single 
 
 3—2 
 
36 MULTIPLY ISOMORPHIC [28 
 
 operation S' of G', while to the operation SpSq there corresponds 
 the operation SpSq, the group G is said to be multiply isomor- 
 phic with the group G'. 
 
 Theorem VII. If a group G is multiply isomorphic with 
 a group G', then (i) the operations of G, which correspond to 
 the identical operation of G',form a self-conjugate sub-group of 
 G ; (ii) to each operation of G' there correspond the same number 
 of operations of G ; and (iii) the order of G is a multiple of the 
 order of G' 
 
 Let So, Si, 02,..., o„_i 
 
 be the set of operations of G which correspond to the identical 
 operation of G'. These operations must form a group, since to 
 SpSq corresponds the operation 1.1, i.e. the identical operation 
 of G' ; and therefore SpSq must belong to the set. 
 
 Again, to the operation T~^SpT of G corresponds the opera- 
 tion J"~' . 1 , T', that is, the identical operation of G'. Hence, 
 whatever operation of G is taken for T, 
 
 T~^ [Sq, *Si,..., Sn-i] T= [So, Si,..., Sn-i]. 
 
 The sub-group T oi G formed of the operations 
 
 is therefore self-conjugate. 
 
 Again, if T and Ti are two operations of G which correspond 
 to the operation T' of G', the operation T~^Ti corresponds to 
 the identical operation of G', and therefore belongs to F. 
 Hence the operations that correspond to T' are all contained 
 in the set TF. The operations of this set are all distinct and 
 equal in number to the order of T. Hence if n is the order of 
 r, to each operation of G' there correspond n operations of G. 
 
 Finally, since to each operation of G there corresponds only 
 one of G', while to each operation of G' there correspond n of G, 
 the order of G' is « times the order of G'. 
 
 To any sub-group of G' of order fi, there corresponds a 
 sub-group of G of order /i7i. For if Tp'Tq forms one of the 
 set 
 
 17" 7' ' rn I rpi 
 
 i, J-i,..., J.p,..., J.q,...^ ± , 
 
29] GROUPS 37 
 
 at least one, and therefore all, of the set TpTqT must occur 
 
 among 
 
 V TV T r Tr TV 
 
 and hence these operations form a group. Moreover, if the 
 sub-group of G' is self-conjugate, so also is the corresponding 
 sub-group of G. 
 
 It should be noticed that no correspondence is thus 
 established between a sub-group of G which does not contain 
 r and any sub-group of G'. 
 
 29. The relation of multiple isomorphism between two 
 groups can be presented in a manner rather different from that 
 of the last paragraph. Let G be any composite group, and 
 r a self-conjugate sub-group of G consisting of the operations 
 
 1/77/77 m 
 
 Then, as in § 22, the operations of G can be arranged in the 
 scheme 
 
 i- , J-i , J-i )■••) ■'■ n—i i 
 
 "m— 1) -'I'Jm— 1> -^ 2"ot— Iv) -'ji— i"m— !• 
 
 Now r being a self-conjugate sub-group of G, it follows that 
 
 and therefore the two sets of operations FSi and SiT coincide 
 except as regards arrangement. Hence 
 
 TaSiTpSj = TaTp'SiSj = TySiSj, 
 
 where 8iT^8i-^ = Tp', 
 
 and T/f^' = Ty; 
 
 so that both Tp- and Ty belong to P. Now all the operations of 
 the set TSiSj occur in a single line, say the (A; -I- l)th, of the 
 above scheme. Hence if any operation of the (i + l)th line be 
 followed by any operation of the (_;-|-l)th line, the result is 
 some operation of the (A;-fl)th line. If then we regard the 
 set of operations contained in each line of the scheme as a 
 
38 FACTOR [29 
 
 single entity, they will by their laws of combination define a 
 new group of order m. In fact, if we denote these entities by 
 
 a relation of the form 
 
 Si'Sj = Sk 
 
 has been proved to hold for every pair. Moreover, these rela- 
 tions necessarily obey the associative law ; for if 
 
 then the relation Sk'Sp = Si'Sq 
 
 follows in consequence of the symbols of G itself obeying the 
 associative law. The symbol So, corresponding to the first line 
 of the scheme, is clearly the symbol of the identical operation in 
 the new group thus defined. 
 
 If now G and T coincide with the groups of the preceding 
 paragraph which are represented by the same symbols, then G' 
 of the preceding paragraph must be simply isomorphic with the 
 group whose operations are 
 
 1, Si, S.,',..., S'm~i- 
 
 It follows that a group G' with which a group G is 
 multiply isomorphic, in such a way that to the identical opera- 
 tion of G' there corresponds a given self-conjugate sub-group 
 r of G, is completely defined (as an abstract group) when G 
 and r are given. This being so it is natural to use a symbol 
 to denote directly the group thus defined in terms of G and 
 r. Herr Holder^ has introduced the symbol 
 
 G 
 
 r 
 
 to represent this group ; he calls it the quotient of G by F, 
 
 and a factor-group of G. We shall in the sequel make constant 
 
 use both of the symbol and of the phrase thus defined. 
 
 C 
 It may not be superfluous to notice that the symbol ^ has 
 
 1 " Zur Ketluotion der algebraischen Cileichmigen," ilath. Ann. xxxiv (1889), 
 p. 31. 
 
30] GROUPS 39 
 
 no meaning', unless T is a self-conjugate sub-group of G. 
 Moreover, it may happen that G has two simply iso- 
 morphic self-conjugate sub-groups F and V. When this is 
 
 the case, there is no necessary relation between the factor- 
 P p 
 
 groups p and p-/ (except of course that their orders are equal) ; 
 
 p 
 in other words, the type of the factor-group =; depends on the 
 
 actual self-conjugate sub-group of G which is chosen for V and 
 not merely on the type of V. 
 
 Further, though in relation to its definition by means of G 
 p ' 
 
 and r we call ^ a factor-group of G, we may without ambiguity, 
 
 since the symbol represents a group of definite type, omit the 
 
 word factor and speak of the group = . 
 
 It is also to be observed that G has not necessarily a 
 
 p 
 sub-group simply isomorphic with ^ . This may or may not be 
 
 the case. 
 
 30. If G is multiply isomorphic with G' so that the self- 
 conjugate sub-group r of (r corresponds to the identical 
 operation of G', it was shewn, at the end of § 28, that to any 
 
 self-conjugate sub-group of G' there corresponds a self-conjugate 
 
 p 
 sub-group of G containing V. Hence, unless ^ is a simple 
 
 group, r cannot be a maximum self-conjugate sub-group of 
 
 C 
 G. If gi is any self-conjugate sub-group of =f , and G^ the 
 
 corresponding (necessarily self-conjugate) sub-group of G, con- 
 
 p 
 taining F, we may form the factor-group ^ , and determine 
 
 again whether this group is simple or composite. By continuing 
 this process a maximum self-conjugate sub-group of G, con- 
 taining F, must at last be reached. 
 
 ' Herr Probenius has extended the use of the symbol to the case ia which r 
 is any group, whether contained in G or not, with which every operation of G is 
 permutable : " Ueber eudliche Gruppen," Berliner Silzimgsberichte, 1895, p. 1G9. 
 We shall always use the symbol in the sense defined in the text. 
 
40 GENERAL ISOMORPHISM [31 
 
 31. Though = is completely defined by Q and T, where F 
 
 is any given self-conjugate sub-group of G, the reader will 
 easily verify that G is not in general determined when T and 
 
 G 
 
 p are given. 
 
 We shall have in the sequel to consider the solution of 
 
 this problem in various particular cases. There is, however, in 
 
 every case one solution of it which is immediately obvious. 
 
 'We may take any two groups (?i and G^, simply isomorphic 
 
 C 
 with the given groups F and =,, such that G^ and G.^ have 
 
 no common operation except identity, while each operation 
 of one is permutable with each operation of the other. The 
 group ((?], G^, formed by combining these two, is clearly such 
 
 that ' ^ ^' is simply isomorphic with y^ \ it therefore gives a 
 
 solution of the problem. 
 
 Definition. If two groups Gi, G^ have no common opera- 
 tion except identity, and if each operation of G^ is permutable 
 with each operation of (rg, the group [G^, G^ is called the 
 direct product of G^ and G«. 
 
 Ex. If H, h are self -conjugate sub-groups of G, and if h is 
 
 contained in //, so that -r- is a self -conjugate sub-group of y , shew 
 
 G H G 
 
 that the quotient of y- by -=- is simply isomorphic with — . 
 
 32. If // is a self-conjugate sub-group of G, of order n, and if 
 
 G G' 
 
 W is a self -conjugate sub-group of (?', of order n', and if — and -jj, 
 
 are simply isomorphic, a correspondence of the most general kind 
 may be established between the operations of G and G'. To every 
 operation of G (or G') there will correspond n (or n) operations of 
 G' (or G), in such a way that to the product of any two operations 
 of G (or <?') there corresponds a definite set of n (or n) operations 
 of G' (or G). Let 
 
 G = H, >%H , S,iH, , S,n-iH, 
 
 and G' = H', S,'H', S^H', , S',n_-,H' ; 
 
 G C" 
 
 and in the simple isomorphism between -^ and -jj, , let <S',. and 8' 
 
33] PEKMUTABLE GROUPS 41 
 
 (r = 0, 1,..., OT-1) be corresponding operations. Then if we take 
 the set S^H' as the n operations of G' that correspond to any 
 operation of the set S,.H of G, and the set S^H as the n operations 
 of G that correspond to any operation of the set (S',.'i/' of G', the 
 correspondence is, in fact, established. 
 
 For, if Aj' and h^ are any two operations of IF, the set of opera- 
 tions S,.'hiSg' ?ij includes it distinct operations only, namely those of 
 the set Sr'SJH'. Hence to the product of any given operation of 
 the set tS,.H by any given operation of the set ,S', H, there corresponds 
 the set of n' operations Sj.'iSjH' ; at the same time the product of the 
 
 two given operations belongs (in consequence of the isomorphism 
 
 G 0'\ 
 
 between ^ and -y^, ) to the set iS^J^^H. The same statements clearly 
 
 hold when we interchange accented and unaccented symbols. 
 
 We still speak of G and G' as isomorphic groups, and the 
 correspondence between their operations is said to give an n-to-n 
 isomorphism of the two groups. We shall return to this general 
 form of isomorphism in dealing with intransitive substitution groups. 
 
 33. Definition. Two groups G and Q' are said to be 
 permutable with each other when the distinct operations of 
 the set SiSj, where for St every operation of the group G is 
 put in turn and for *Si,' every operation of the group G', coincide 
 with the distinct operations of the set S/Si except possibly 
 as regards arrangement. 
 
 If the two groups G and G' are permutable, the group 
 {G, G'] must be of iinite order. For, by the definition, every 
 operation 
 
 ...SpSq'SrSt'... 
 
 can be reduced to the form SiSj ; and therefore the number of 
 distinct operations of the group {G, G'} cannot exceed the 
 product of the orders of G and G'. Let g be the group formed 
 of the common operations of G and G'. Divide the operations 
 of these groups into the sets 
 
 g, %g , 2,g,..., tm-ig 
 
 and g, gXi, gl.,', , gt'^'-i- 
 
 Then every operation of the set SiSj can clearly be ex- 
 pressed in the form 
 
42 PERMUTABLE [33 
 
 where 7 is some operation of g. And no two operations of this 
 form can be identical, for if 
 
 2^71^9' = 2r722/, 
 
 then 72-^2, .-^SpYi = 2/ S,,'"' ; 
 
 so that S/Sj'"' belongs to g. But this is only possible if 
 
 which leads to 2^ = 2^, 
 
 and 72 = 7i • 
 
 The order of [G, G'] is therefore the product of the orders of G 
 
 and G', divided by the order of g. 
 
 If every operation of G is permutable with G', then g must 
 be a self-conjugate sub-group of G. For G and G' are trans- 
 formed, each into itself, by any operation of G ; and therefore 
 their common sub-group g must be transformed into itself by 
 any operation of G. 
 
 Moreover those operations of G, which are permutable with 
 every operation of G', form a self-conjugate sub-group of G. 
 For if T is an operation of G, which is permutable with every 
 operation 8' of G', so that 
 
 T-'8'T=S', 
 
 and if S is any operation of G, then 
 
 S"T-'S.S-^S'S . S-'TS = S-'S'S, 
 
 so that 8~^TS is permutable with every operation of G'. Hence 
 every operation of G, which is conjugate to T, is permutable 
 with every operation of G' ; and the operations of G, which are 
 permutable with every operation of G', therefore form a self- 
 conjugate sub-group. 
 
 If G is a simple group, g must consist of the identical 
 operation only ; and either all the operations of G, or none of 
 them, must be permutable with every operation of G'. 
 
 A special case is that in which the two groups G' and G are 
 respectively a self-conjugate sub-group F and any sub-group H 
 of some third group; for then every operation of H is per- 
 mutable with r. If if is a cyclical sub-group generated by an 
 operation 8 of order n, and if 8™ is the lowest power of 8 
 
34] GROUPS 4.3 
 
 which occurs in F, then m must be a factor of n. For if m is 
 the' greatest common factor of m and ?i, integers *■ and y can be 
 found such that 
 
 ccm + yn = m. 
 
 Now S'""' = S^'"''*''-'^ = >S™' 
 
 and therefore /S'"'' belongs to F. Hence m' cannot be less than 
 m, and therefore m is a factor of n. Moreover, since {S™} is 
 
 a sub-group of F, the order of F must be divisible by — . 
 
 Hence : — 
 
 Theorem VIII. If an operation 8, of order n, is permutable 
 with a group F, and if S"' is the lowest power of S which occurs 
 
 in F ; then m is a factor of n, and — is a factor of the order 
 
 ofT. 
 
 The operations of (F, 8] can clearly be distributed in the 
 sets 
 
 F, TS, F6^..., F6"™-^; 
 
 and no two of the operations S, S^,..., *S'™~^ are conjugate in 
 
 (r, >sf}. 
 
 34. A still more special case, but it is most important, is 
 that in which the two groups are both of them self-conjugate 
 sub-groups of some third group. If in this case the two groups 
 are G and H, while 8 and T are any operations of the two 
 groups respectively, then 
 
 8-'H8==H, 
 
 and T-'GT=G; 
 
 so that every operation of G is permutable with H and every 
 operation of H is permutable with G. 
 
 Consider now the operation 8-^T-'8T. Regarded as the 
 product of 8-'' and T^hST it belongs to G, and regarded as the 
 product of /Sf-^r-'S and T it belongs to H. Every operation of 
 this form therefore belongs to the common group of G and H. 
 If G and H have no common operation except identity, then 
 
 8-'T-'8T=l, 
 
44 EXAMPLES [34 
 
 or 8T=TS; 
 
 and S and T are permutable. Hence : — 
 
 Theorem IX' If every operation of G transforms H into 
 itself and every operation of H transforms G into itself, and 
 if G and H have no common operation except identity ; then 
 every operation of G is permutahle with every operation of H. 
 
 Corollary. If every operation of G transforms H into itself 
 and every operation of H transforms G into itself, and if either 
 G or if is a simple group ; then G and H have no common 
 operation except identity, and every operation of G is permutable 
 with every operation of H. 
 
 For, by § 33, if G and H had a common sub-group, it would 
 be a self-conjugate sub-group of both of them ; and neither of 
 them could then be simple, contrary to hypothesis. Conse- 
 queatly, the only sub-group common to G and H is the 
 identical operation. 
 
 35. If N and N' are the orders of two permutable groups G and 
 G' , the NN' equations expressing every operation of the form SS', 
 where S belongs to G and S' to G' , in the form 2' 2, where 2 and 2' 
 belong to G and G' respectively, are never all independent. In 
 particular, if 
 
 are a set of independent generating operations of G, and 
 
 wij, nio, , ni^ 
 
 their orders, it is clearly sufficient that each operation of the form 
 
 -S",?/ (a=l, 2, ,mp-l;p = l, 2, , r) 
 
 should be capable of expression in the form 22'. 
 
 When, in fact, these conditions are satisfied, it is always possible 
 by a series of steps to express any operation iS"<b' in the form 22'. 
 
 Ex. 1. Shew that, in the group whose defining relations are 
 
 ^1^ = 1, B'=l, {ABr = l, 
 
 the three operations A^, B~^A''B, BA'-B~' are permutable and that 
 they form a complete set of conjugate operations. Hence shew that 
 {.4-, B} is a self-conjugate sub-group, and that the order of the group 
 
 is 24. 
 
 ' Dyck, " Gruppentheoretisclie Studien,'' Matli. Ami. Vol. xxii (1883), p. 97. 
 
35] EXAMPLES 45 
 
 Ex. 2. Shew that the cyclical group generated by the substitu- 
 tion (1234567) is permutable with the group 
 
 {(267) (345), (23) (47)}; 
 
 and that the order of the group resulting from their product is 168. 
 
 Ex. 3. If g^ and c/., are the orders of the groups G^ and G«, y 
 the order of their greatest common sub-group and g the order of 
 {G'], (?,,}, shew that 
 
 ffy > ffiff«, 
 and that, if gy -gig.;,, then G^ and C, are permutable. (Frobenius.) 
 
 Ex. 4. If (?i and G^ are two sub-groups of G of orders g^ and g„, 
 and S any operation of G, prove that the number of distinct opera- 
 tions of G contained in the set S^SSr., when for S-^ and S„ are put in 
 
 turn everj' pair of operations of G^ and G., respectively, is '-- ; y 
 
 being the order of the greatest sub-group common to *S~' G-^S and G^. 
 If T is any other operation of G. shew also that the sets S-^SS^ 
 and SjTS., are either identical or have no operation in common. 
 
 (Frobenius.) 
 
 Ex. 5. If a group G of order nin has a sub-group H of order n, 
 and if n has no prime factor which is less than m, shew that // 
 must be a self -conjugate sub-group. (Frobenius.) 
 
CHAPTER IV 
 
 ON ABELIAN GROUPS. 
 
 36. We shall now apply the general results, that have 
 been obtained in the last chapter, to the study of two special 
 classes of groups ; in the present chapter we shall deal par- 
 ticularly with those groups whose operations are all permutable 
 with each other. 
 
 Definition. A group, whose operations are all permutable 
 with each other, is called an Abelian^ group. 
 
 It is to be expected (and it will be found) that the theory of 
 Abelian groups is much simpler than that of groups in general ; 
 for the process of multiplication of the operations of such groups 
 is commutative as well as associative. 
 
 Every sub-group of an Abelian group is itself an Abelian 
 group, since its operations are necessarily all permutable. For 
 
 ' On Abelian groups, the reader may consult Probenius and Stickelberger, 
 " Ueber Gruppen vertausohbarer Elemente," Crelle's Jownal, Yd. lxxxvi (1879), 
 p. 217 ; and a very complete discussion in the second volume of Herr Weber's 
 recently published Lehrbuch der Algebra. In the proof of the existence of a 
 set of independent generating operations (§ 41) we have directly followed Herr 
 Weber. 
 
 The name "Abelian group" has been applied by M. Jordan {Traite des sub- 
 stitutions etc. pp. 171 et seq.) to an entirely different class of groups, whose 
 operations are not permutable. Most writers, we believe, have used the phrase 
 in the sense defined in the text. 
 
 The connection of Abel's name with groups of permutable operations is due 
 to his having been the first to investigate, with complete generality, the applica- 
 tion of such groups to the theory of equations, "M^moire sur une olasse 
 particuMre d'fiquations r^solubles alg^briquement," Crelle's Journal, Vol. iv 
 (1829), p. 131 ; or Collected Works, 1881 edition, Vol. i, p. 478. 
 
37] GENERAL PROPERTIES OF ABELIAN GROUPS 47 
 
 the same irasun, every operation and every sub-group of an 
 Abelian group is self-conjugate both in the gi'oup itself and in 
 any sub-group in which it is contained. 
 
 If is an Abelian group and H any sub-group of 0, then 
 since H is necessarily self-conjugate, there exists a factor- 
 
 C 
 group -jf, and this again must be an Abelian group. '(The 
 
 reader must not however infer that, if H and -^ are both 
 
 Abelian, then G is also Abelian. It is indeed clear that this 
 is not necessarily the case.) 
 
 37. Let now G be any Abelian group, and let p^ be the 
 highest power of a prime p that divides its order. We shall 
 first shew that G has a single sub-group of order p^, consisting 
 of all the operations of G whose orders are powers of p. 
 
 If ;Si, 5^2 are any two operations of G, it follows from § 33, 
 
 that, because /Si and 8^ are permutable, the order of {>S'i, (S'aj is 
 
 equal to or is a factor of the product of the orders of S^ and S^. 
 
 So again, if Ss is an operation of G not contained in [8i, S^, the 
 
 order of [S^, S^, S^] is equal to or is a factor of the product of 
 
 the orders of {S-,, S^} and Sg. Now by continually including a 
 
 fresh operation, not contained in the group already arrived at, 
 
 we must in this way after a finite number of steps arrive at the 
 
 group G, whose order is divisible by p. Hence, among the 
 
 operations Si, S^, 8s, , there must be at least one whose 
 
 order is divisible by p, and some power of this, say 8, will be 
 
 an operation of order p. Now, if m is greater than unity, the 
 
 O 
 order of the factor-group j-™ , which is also Abelian, is divisible 
 
 by p, and therefore this factor-group must have an operation of 
 order p. Hence G will (§ 28) contain a sub-group of order 
 p\ If m is greater than 2, the same reasoning may be 
 repeated to shew that G has a sub-group of order p^, and so on. 
 Hence, finally, G has a sub-group of order p'^. Let H be this 
 sub-group ; and suppose if possible that G contains an operation 
 2\ whose order is a power of p, which does not belong to H. 
 Then {H, T] is a, sub-group of G, whose order is a power of p, 
 greater than p™ ; and this is impossible (§ 22). The sub-group 
 
48 GENERAL PROPERTIES OF ABELIAN GROUPS [37 
 
 H must therefore contain all the operations of G whose orders 
 are powers of p. Hence : — 
 
 Theorem I. If p™ is the highest power of a prime p that 
 divides the order of an Abelian group G, then G contains a 
 single suh-group of order p^'\ which consists of all the operations 
 of G whose orders are powers of p. 
 
 38. Let the order of G be 
 
 N=p,'"'p,"'' ?V"", 
 
 where jOj , p.^, , pn are distinct primes ; and let 
 
 -ffi, H-2, , Ifn 
 
 be the sub-groups of G of orders 
 
 Pi ) Pi ) ) Pn 
 
 Since the orders of H^ and H^ are relatively prime, they 
 can have no common operation except identity ; and therefore 
 the order of {^i, H^] is pi"^p.,"'K This sub-group contains all 
 the operations of G whose orders are relatively prime to 
 
 N 
 —^^ — — . For if T were an operation of G of order p^^p^, not 
 Pi Pi 
 
 contained in {^i, H„], then {H^, H.,, T\ would be a sub-group 
 of G, of order pi"'p."= , where n^ > m^ if a > and n., > m^ if j8 > ; 
 and this is impossible (§ 22). 
 
 This process may clearly be continued to shew that, if 
 N = /jLv, where p, and v are relatively prime, then G contains a 
 single sub-group of order fi, consisting of all the operations of 
 G whose orders are relatively prime to v. Moreover G itself is 
 the direct product (§ 31) of IT^, H^, , J7„. 
 
 39. The first problem of pure group-theory that presents 
 itself in connection with Abelian groups is the determination 
 of all distinct Abelian groups of given order N. Let Hi 
 and Hi' be two distinct Abelian groups of order pi"\ i.e. two 
 groups which are not simply isomorphic. Then two Abelian 
 groups of order N, whose sub-groups of order pi"' are simply 
 isomorphic with H and Hi respectively, are necessarily distinct. 
 
 Since then G is the direct product of H, H^ , H^, the 
 
 general problem for any composite order N will be completely 
 solved when we have determined all distinct types of Abelian 
 
40] 
 
 INDEPENDENT GENERATING OPERATIONS 
 
 49 
 
 groups of order p^\ jJs^s , Pn™"- We may therefore, for the 
 
 purpose of this problem, confine our attention to those Abelian 
 groups whose orders are powers of primes. 
 
 40. Suppose then that Q is an Abelian group whose order 
 is the power of a prime. If among the operations of we 
 choose at random a set 
 
 P, P', P" 
 
 from which the group can be generated, they will not in 
 general be independent of each other. 
 
 As an instance, we may take the group whose multiplication 
 table is : — 
 
 1 A P. P. A A P. Pr 
 
 1 
 P. 
 
 P. 
 P. 
 
 P. 
 Pr 
 
 Here /"^ and F^ are two operations from which every operation 
 of the group may be generated : an inspection of the table will 
 shew that they are connected by the relation 
 
 On the other hand, if we choose P^ and P^ as generating 
 operations, we find that every operation of the group can be ex- 
 pressed in the form 
 
 P,"P/, (a = 0, 1,2, 3; ^ = 0,1), 
 
 while the only conditions to which the permutable operations P^ 
 and P4 are submitted are 
 
 P,*^.l, P/ = l. 
 
 The question then arises as to whether G can be generated 
 by a set of permutable and independent operations, i.e. by a 
 set of permutable operations which are connected by no 
 relations except those that give their orders. That this 
 
 1 
 
 Pi 
 
 P. 
 
 Ps 
 
 P4 
 
 A 
 
 P. 
 
 Pr 
 
 Pi 
 
 A 
 
 P. 
 
 1 
 
 P. 
 
 A 
 
 Pr 
 
 A 
 
 p. 
 
 Ps 
 
 1 
 
 Pi 
 
 P. 
 
 Pr 
 
 A 
 
 A 
 
 A 
 
 1 
 
 Pi 
 
 P. 
 
 Pr 
 
 P. 
 
 A 
 
 P. 
 
 P. 
 
 P. 
 
 A 
 
 Pr 
 
 1 
 
 Pi 
 
 A 
 
 Ps 
 
 A 
 
 P. 
 
 Pr 
 
 P. 
 
 Pi 
 
 A 
 
 Ps 
 
 1 
 
 Pe 
 
 Pr 
 
 P. 
 
 P. 
 
 P. 
 
 Ps 
 
 1 
 
 Pi 
 
 Pr 
 
 A 
 
 A 
 
 Pe 
 
 Ps 
 
 1 
 
 Pi 
 
 P. 
 
50 INDEPENDENT GENERATING OPERATIONS [41 
 
 question is always to be answered in the affirmative may be 
 proved in the following manner. 
 
 41. Let the order of Ghe p'^; and let Pj be an operation 
 of G whose order ^j™' is not less than that of any other opera- 
 tion of the group. Then every operation of the group satisfies 
 the equation 
 
 If mi = m, the order of {Pi} is equal to the order of ; the 
 latter is then a cyclical group generated by the operation Pj. 
 
 If mi < m, G must contain other operations besides those of 
 {Pi}. Denoting {Pi} by G^, let Q be any operation of not 
 contained in G,, and let Q" be the lowest power of Q that is 
 contained in Gi. Then (§ 33) a must be a power of p ; and 
 when for Q each operation of G that is not contained in Gi is 
 taken in turn, a must have some maximum value, say p'^. 
 Since no operation of G is of greater order than p*"', it follows 
 that m^ ^ nil. We may suppose then Q to be an operation of 
 G, such that Qp"^ is the lowest power of Q which is contained 
 in(?i. 
 
 Then Qj''»» = P,^, 
 
 and Qr.=l = Pj^«'".-'»., 
 
 so that Xi is divisible by p'^ and may be expressed in the form 
 ocp™^. The case Xi = forms no exception to this statement, 
 since Xi is congruent to zero, mod. p"^^; but we actually take 
 a; = in this case. 
 
 If now we write QPi~* = P2, 
 
 then P,P'''=l, 
 
 and P^P"' is the lowest power of Pj which is contained in Oi. 
 Let the sub-group {Pi, Qj be denoted by G^. Then G^ is 
 generated by the two independent operations Pi and P, of 
 orders p™' and p™^ ; and every operation S of G is such that 
 
 where Mi is divisible by p™^ 
 
 This process may now be continued step by step. That 
 it will ultimately lead to a representation of G as generated 
 by a set of independent operations may be shewn by induction. 
 
41] OF ABELIAN GROUPS 51 
 
 To this end, we will suppose that a sub-group On has been 
 arrived at which is generated by the n independent operations 
 P P P 
 
 of orders ^j™', p™% , p"^ , 
 
 where mi>m2> >m,i. 
 
 We will also suppose that, if T is any operation of G, 
 
 T'pm,, ^ p oLiP a2 pa.,,-! 
 
 ■^ -^ 1 -^ 2 -^ n-1' 
 
 where a.i, ok, , On-i are all divisible by^™". In the special 
 
 case re = 2 these suppositions have been justified. 
 
 Let Q' be any operation of Q, and let Q'" be the lowest 
 power of Q' that occurs in On. Then when for Q' each operation 
 of that is not contained in On is taken in turn, a will have 
 a certain maximum value, say p'"»+i ; and from the suppositions 
 made above, m„+i is not greater than to„. We may therefore 
 write 
 
 Q>™„+1 = P^^^Pi^ Pt\Pn^": 
 
 and hence 
 
 Q'p^n — p PiP^n -'»»+ip P^m„ -m„4., pP,,-lpm„ -m^^^p /S„i)">„ -m„+i^ 
 
 Now Pi, P2, , Pn are independent; it follows there- 
 fore from the second assumption made above, that the 
 
 exponents of Pj, Pj, , Pn in this last equation are divisible 
 
 byp""; and hence that A) /^a, , /3ji are divisible by^J""*'. 
 
 If we now suppose that Q' itself is chosen so that Q'p'""+^ is 
 the lowest power of Q' that occurs in On, then 
 
 Q'pm„+l_pXiP™n+ipX2P™>i+l _p Ohtp^rt+l. 
 
 hence, if QTr'^'Pf^^ P«-*» = Pn+i, 
 
 then PP"'"+' = 1 ■ 
 
 and P^^^ is the lowest power of P that occurs in On- Hence 
 finally (Pi, P^, , P„, Q'} is generated by the n + 1 inde- 
 pendent operations Pj, P^, , P„+i of orders p™', p"^\ , 
 
 ^m«+i (,^j ^ TOj > > nin+i) ; and S being any operation of 0, 
 
 8P''"+' = P,y^P,n Pnf", 
 
 where 71, 72, > 7n are divisible by p^+K 
 
 4—2 
 
52 INDEPENDENT GENERATING OPERATIONS [41 
 
 This completes the proof by induction, and we may now 
 state the result in the form of the following theorem : — 
 
 Theorem II. An Aheliam. group of order p^, where p is a 
 prime, can be generated by r(< m) independent operations P^, 
 
 Pa, , Pr, of orders p™', p"^, , p'^', where 
 
 mi + m2+ ... + m,. = TO, 
 
 and TO, > TO2 > ^ TO^. Moreover if fjL is any positive integer 
 
 such that rus'^fi'^ w,«+i. while Q is any operation of the group, then 
 qp^^pa^p^a, P-., 
 
 where oti, a^, , ots are divisible by p"'. 
 
 The actual existence of the set of independent generating 
 operations is demonstrated by the above inductive proof. 
 The other inferences in the theorem may be established as 
 follows. A set of independent permutable operations of orders 
 
 p™i, p'^\ , p^' generate a group of order ^™i+'%+-.+™r^ and 
 
 therefore 
 
 TOi + iria + + TOy = TO. 
 
 Since each number in this equation is a positive integer, 
 the number of terms on the left-hand side cannot be greater 
 than m. Hence 
 
 r'^m; 
 and, if r = TO., every operation of the group except identity is of 
 order p. 
 
 Finally, from the above inductive proof it follows that, if Q 
 is any operation of the group, then 
 
 Qp'^-+' = P/^P/^ p^3,_ 
 
 where ySj, ySa, , 0s are all divisible by p™'+i. 
 
 Hence Qp" = p/>y''-'»'+ip/.p''-""« p/ pi'-"'-+' 
 
 = p,-''P,-'^ Pa 
 
 where Ui, a^, , as are all divisible by p^*. 
 
 42. It is clear, from the synthetic process by which it has 
 been proved that an Abelian group of order p"^ can be generated 
 by a set of independent operations, that a considerable latitude 
 exists in the choice of the actual generating operations ; and 
 the question arises as to the relations between the orders of the 
 distinct sets of independent generating operations. 
 
 The discussion of this question is facilitated by a considera- 
 tion of certain special sub-groups of G. If A and B are two 
 
43] OF ABELTAN GROUPS 53 
 
 operations of G, and if the order of A is not less than that of B, 
 the order of AB is equal to, or is a factor of, the order of A. 
 Hence the totality of those operations of G whose orders do not 
 exceed p*^, or in other words of those operations which satisfy 
 the relation 
 
 8^ = 1, 
 form a sub-group G^. The order of G^^ clearly depends on the 
 orders of the various operations of G and in no way on a special 
 choice of generating operations. Now if 
 
 P-,"-^P^"^ Pa 
 
 belongs to G^, then 
 
 p^a^pl^ p^a^pl^ P/rJ''* = 1. 
 
 Hence if mj+i is the first of the series 
 
 nil, mg, , m,., 
 
 which is less than fi, then as+i, , a^ may have any values 
 
 whatever; but a^ (^ = 1, 2, , s) must be a multiple of p"^~i^. 
 
 It follows from this that G^ is generated by the r indepen- 
 dent operations 
 
 ppitii-ii. pj/in^-ii p pm,-i^ p p 
 
 If then the order of G^ is p", we have 
 
 r 
 
 V = fis+ "2 mt. 
 
 s+1 
 
 The order of Gi, the sub-group formed of all operations of G 
 whose order is p, is clearly p'^ 
 
 43. Suppose now that by a fresh choice of independent 
 generating operations, it were found that G could be generated 
 by the r' independent operations 
 
 p> p' p' 
 
 of orders p™'', p'^"', , p™ ', 
 
 where mi' > m/ ^ > m^. 
 
 If m's+i is the first of this series which is less than fi, the 
 order of G^ will be p"', where 
 
 r' 
 
 v' = /j-s' + 2 mt . 
 
 s'+l 
 
 The order of G^ is independent of the choice of generating 
 operations ; so that for all values of /u. 
 
 V = v. 
 
54 DISTINCT TYPES [43 
 
 Hence, by taking /a = 1, 
 
 r = r', 
 
 or the number of independent generating operations is indepen- 
 dent of their choice. 
 
 If now irit = mt' (t = s-\- 1, s + 2, ,r), 
 
 and nig > m/, 
 
 and if we choose /^ so that 
 
 mj ^ yu. > m/ ; 
 
 r 
 
 then v = fjLS+ 2 mj , 
 
 s+l 
 
 and z/' = /u, (s — 1) + 771/ + 2 mj. 
 
 s+l 
 
 The condition 1/ = ly' 
 
 gives yu, = mg', 
 
 in contradiction to the assumption just made. 
 
 Similarly we can prove that the assumption nis > nig cannot 
 be maintained; hence 
 
 m, = nig ; 
 and therefore, however the independent generating operations 
 of G are chosen, their number is always r, and their orders are 
 
 f ' f > ' i' • 
 
 44. If G' is a second Abelian group of order ^™, simply 
 isomorphic with G, and if 
 
 P' P' P ■ 
 
 of orders p'^'', p^"', , p™'', 
 
 where m/ > m^' > > ra/, 
 
 are a set of independent generating operations of G', exactly the 
 same process as that of the last paragraph may be used to shew 
 that 
 
 r = r', 
 
 and mj = ?7i/(s = 1, 2, r). 
 
 In fact, since corresponding operations of two simply iso- 
 morphic groups have the same order, the order of G^ must be 
 equal to the order of G/ ; and this is the condition that has 
 been used to obtain the result of the last paragraph. 
 
 Two Abelian groups of order p™ cannot therefore be simply 
 
46] OP ABELIAN GROUPS 55 
 
 isomorphic unless the series of integers mi, 7112, , m,. is the 
 
 same for each. On the other hand when this condition is satis- 
 fied, it is clear that the two groups are simply isomorphic, since 
 by taking Pg and P/ (s= 1, 2, , j^) as corresponding opera- 
 tions, the isomorphism is actually established. 
 
 The number of distinct types of Abelian groups of order 
 p™, where p is a prime, i.e. the number of such groups no one 
 of which is simply isomorphic with any other, is therefore equal 
 to the number of partitions of m. When the prime p is given, 
 each type of group may be conveniently, and without ambiguity, 
 represented by the symbol of the corresponding partition. 
 Thus the typical group G that we have been dealing with 
 would be represented by the symbol (mj, m^, , m^). 
 
 45. Having thus determined all distinct types of Abelian 
 groups of order p™, a second general problem in this connec- 
 tion is the determination of all possible types of sub-group 
 when the group itself is given. This will be facilitated by the 
 consideration of a second special class of sub-groups in addition 
 to the sub-groups (z^ already dealt with. 
 
 If S and >Si' are any two operations of G, then 
 
 and therefore the totality of the distinct operations obtained by 
 raising every operation of G to the power pf^ will form a 
 sub-group H^. 
 
 If ms'^fi>ms+i, 
 
 then (Theorem II, § 41) 
 
 Sp" = P^'^^P^P^'^^P'' P/.J"", 
 
 8 being any operation of G. Hence H^ is generated by the s 
 independent operations 
 
 p vi^ p pV- P pi^- 
 
 a 
 
 and the order of H,^ is p^ ' . 
 
 46. Let now T of type (mi, Wj, , ««) be any sub-group of 
 
 G. The order of the group Tj, formed of all the operations of 
 r which satisfy the equation 
 
56 SUB-GROUPS [46 
 
 is p" (§ 42). This group must be identical with or be a sub- 
 group of G^, whose order is p'- Hence 
 
 i.e. the number of independent generating operations of any 
 sub-group of G is equal to or is less than the number of inde- 
 pendent generating operations of G itself. 
 
 If p"- is the order of F, there must be one or more sub- 
 
 P 
 
 groups of G of order p"+i which contain V; for -=f has operations 
 
 of order p. Hence V must be contained in one or more 
 sub-groups of G of order p'^^. We may begin then by 
 considering all possible types of sub-groups of G of order p™-' 
 
 Let g be such a sub-group and (m/, m^, ,mV) its type; and 
 
 suppose that 
 
 mt = m.t'(t= 1, 2, , k-l), 
 
 and mj =)= »^«;'- 
 
 Then if mj < m^', 
 
 the order of h,n^, the sub-group of g which results by raising all 
 its operations to the power p^^, is greater than the order of Hm . 
 This is impossible, since hm must coincide with 5^ or with one 
 of its sub-groups. Hence 
 
 mil! < ink- 
 
 r' r 
 
 Now Xmt = 2wit — 1 ; 
 
 1 1 
 
 therefore m^.' = mj. — 1, 
 
 and nit =mt(t = k + 1, i^)- 
 
 In the particular case in which m^ is unity if we take k 
 equal to r, my is zero ; and in this case g would have r - 1 
 generating operations. 
 
 It is easy to see that sub-groups of order p™~^ and of all 
 the types just determined actually exist. For 
 
 "it -^ 2> > "h-\> ^]?> "k+i> ' -^ >■ 
 
 are a set of independent operations which generate a sub-group 
 of the type (mi, m^,..., mk--,, nin-l, rrik+x,..., tUr). 
 
47] OF ABELIAN GROUPS 57 
 
 47. Returning to the general case, we will assume for all 
 
 orders not less than p" the existence of a sub-group V of type 
 
 (rii, n^, , n,), when the inequalities 
 
 t t 
 
 '2,n„^'Zmu(t= 1, 2, r) 
 
 1 1 
 
 are satisfied, where if ^ > s, 7i< is zero. When n is equal to m— 1, 
 the truth of this assumption has been established. 
 
 If r' of type (til, 71.2, , wV) is a sub-group of order p"'~^, it 
 
 must be contained in some sub-group of order p^, say F of type 
 («!, n.,, , ris). Then, by the preceding discussion, 
 
 t t 
 
 Sw„'<Sw„(*=l, 2, ,s); 
 
 1 1 
 
 t t 
 
 and therefore Swu' < Sm„ {t=l, 2, , r), 
 
 1 1 
 
 so that the conditions assumed for sub-groups of order not less 
 
 than p^ are necessary for sub-groups of order p^~^. Moreover if 
 
 the conditions 
 
 t t 
 
 2n„' < Sm„ {t = l,'2, ,r) 
 
 1 1 
 
 are satisfied, and if %' is the first of the series 
 
 n^ , % ....... , 
 
 which is less than the corresponding term of the series 
 
 we may take n„ = w^', u^h, 
 
 % = Wfc' + 1 ; 
 
 and then the conditions 
 
 t t 
 
 2n«<Sm„(i=l, 2, , r) 
 
 1 1 
 
 are satisfied. But these conditions being satisfied it follows, 
 
 from the assumption made, that G contains a sub-group of order 
 
 p^ and type (wj, n^, n^ and therefore also a sub-group of 
 
 order p^-^ and type {n^, n^, , n^). If the conditions assumed 
 
 are sufficient for the existence of a sub-group of order p^, they 
 
 are thus proved to be sufficient for the existence of one of 
 
 order p'^^. Hence since they are sufficient in the case of 
 
 sub-groups of order p^~^, they are so generally. 
 
58 ABELIAN GROUPS WHOSE OPERATIONS [47 
 
 We may summarize the results obtained iu the last three 
 paragraphs as follows : — 
 
 Theorem III. The number of distinct types of Abelian 
 groups of order p^, where p is a prime, is equal to the number of 
 partitions of m; and each type may he completely represented 
 
 by the symbol (mj, m^ , m,.) of the corresponding partition. 
 
 If the numbers in the partition are written in descending order, 
 
 a group of type (mi, m^,, , m^) will have a sub-group of type 
 
 (ill, 1^2 1 ) "s). when the conditions 
 
 s^r, 
 t t 
 
 2n„ :$ 2m„ (i = 1, 2, ,r), 
 
 1 1 
 
 are satisfied ; and the type of every sub-group must satisfy these 
 conditions. 
 
 48. It will be seen later that the Abelian group of order 
 
 p™ and type (1, 1, 1, , with m units) is of special importance 
 
 in the general theory, and we shall here discuss one or two of 
 its simpler properties. 
 
 Since the generating operations of the group are all of order 
 p, every operation except identity is of order p ; and therefore 
 
 the type of any sub-group. of order p' is (1, 1, 1, with s 
 
 units). If the group be denoted by 0, every sub-group (r^ 
 coincides with ; while of the sub-groups H^, the first coincides 
 with G and all the rest consist of the identical operation only. 
 
 In choosing a set of independent generating operations, we 
 may take for the first, Pj, any one of the p™— 1 operations of 
 the group, other than identity. The sub-group {PJ is of order 
 p ; and therefore has p™ —p operations which are not contained 
 in {Pi}. If we choose any one of these, Pa, it is necessarily 
 independent of Pj, and may be taken as a second generating 
 operation. The sub-group {Pi, P^ is of order p"^ and type 
 (1, 1) ; and G has p^ —p^ operations which are not contained in 
 this sub-group. If Pg be any one of these, no power of Ps 
 other than identity is contained in {Pj, Pa} ; and Pi, Pa, Ps are 
 therefore three independent operations which generate a sub- 
 group of order _p'. This process may clearly be continued till 
 all m generating operations have been chosen. If then the 
 
49] ARE ALL OF PRIME ORDER 59 
 
 position which each generating operation occupies in the set of 
 m, when they are written in order, be taken into account, there are 
 
 (i3™-l)(p™-_p)(p™-^2) (p^-p™-!) 
 
 distinct ways in which a set may be chosen. If on the other 
 hand the sets of generating operations which consist of the 
 same operations written in different orders be regarded as 
 identical, the number of distinct sets is 
 
 m! 
 
 49. No operation P of the group can belong to two distinct 
 
 sub-groups of order p except the identical operation. Hence 
 
 since every sub-group of order p contains p—1 operations 
 
 M™ — 1 
 besides identity, Q must contain ^ — -y sub-groups of order p. 
 
 Let Nm, r he the number of sub-groups of of order p^, so 
 
 that 
 
 «™ — 1 
 AT =iL 
 
 ^V> p-1- 
 There are, in 0, p^—p^ operations not contained in any given 
 sub-group of order p^. If P occurs among these operations, so 
 
 also do P^, P^,...,. , PP~^. Hence there are ^ sub-groups of 
 
 order p in G which are not contained in a given sub-group of 
 order p^. Each of these may be combined with the given 
 sub-group to give a sub-group of order p'+\ When every 
 sub-group of order p^ is treated in this way, every sub-group of 
 order p^+^ will be formed and each of them the same number, x, 
 of times. Hence 
 
 •*'-'■' m, r+i -'-'m,r „ _ i " 
 p 1 
 
 Now a sub-group of order p''+^ contains Nr+i^r sub-groups of 
 
 order p'', and r^ sub-groups of order p which are not 
 
 contained in any given sub-group of order p^. Hence 
 
 p-1 
 
 and therefore iV, 
 
 '"■'■+^~i\r,+,/ p-1 ■ 
 
60 EXAMPLES [49 
 
 We will now assume that 
 
 „ _ (p^-l)(p'^-'-l) (p'^'+i-l) 
 
 ^'"■*- (^_l)(p=_l) (pt-i) ' 
 
 for all values of m and for values of t not exceeding r. This 
 has been proved for r = 1. Then it follows, from the above 
 relation, that 
 
 _ (p-^-l)(p-^-'-l) ip^-l) 
 
 ''"'■'•+^ (p-l){p'-l) (p'^+^-1) ' 
 
 that is to say, if the result is true for values of t not exceeding 
 r, it is also true when t = r + l. Hence the formula is true 
 generally. 
 
 It may be noticed that 
 
 ■"m,t ^ ■'^ m,m,—t- 
 
 50. Ex. 1. Shew that a group whose operations except 
 identity are all of order 2 is necessarily an Abelian group. 
 
 Ex. 2. Prove that in a group of order 16, whose operations 
 except identity are all of order 2, the 15 operations of order 2 may 
 be divided into 5 sets of 3 each so that each set of 3 with identity 
 forms a sub-group of order 4 ; and that this division into sets may 
 be carried out in 7 distinct ways. 
 
 Ex. 3. If G is an Abelian group and H a sub-group of G, 
 shew that G contains one or more sub-groups simply isomorphic 
 
 with -=i. 
 a 
 
 Ex. 4. If the symbols in the successive rows of a determinant 
 of n rows are derived from those of the first row by performing on 
 them the substitutions of a regular Abelian group of order n, prove 
 that the determinant is the product of n linear factors. 
 
 {Messenger of Mathematics, Vol. xxiii p. 112.) 
 
 Ex. 5. Discuss the number of ways in which a set of 
 independent generating operations of an Abelian group of order jo™ 
 and given type may be chosen. Shew that, for a group of type 
 
 (mi, m^, , m,.), where in^>m^> > m„ the number of ways is 
 
 of the form p'^ {p— 1)''; and in particular that for a group of order 
 
 piHn+i) and type (n, n — 1, , 2, 1), the number of ways is 
 
 ^"(^—1)", where v= ^Ji (w -I- 1) (2n -I- 1) — ji. 
 
 Ex. 6. Shew that for any Abelian group a set of independent 
 generating operations 
 
 Oi, 02,..., Or-l) Orvj "it 
 
 can be chosen such that, for each value of r, the order of Sr is equal 
 to, or is a factor, of the order of <S',._i. 
 
CHAPTER V. 
 
 ON GROUPS WHOSE ORDERS ARE THE POWERS OP 
 PRIMES. 
 
 51. Having in the last chapter dealt in some detail with 
 Abelian groups of order p™, where ^ is a prime, we shall now in- 
 vestigate some of the more important properties of groups, which 
 have the power of a prime for their order but are not necessarily 
 Abelian. Besides illustrating and leading to many interesting 
 applications of the general theorems of Chapter iii, the dis- 
 cussion of gi'oups, whose order is the power of a prime, will be 
 found in many ways to facilitate the subsequent discussion of 
 other groups, whose order is not thus limited. 
 
 52. If G' is a group whose order is p^, where p is a prime, 
 the order of every sub-group of G must also be a power of p; and 
 therefore (§ 2.5) the total number of operations of G which are 
 conjugate with any given operation must be a power of ^. The 
 identical operation of G is self-conjugate. Hence if the opera- 
 tions of G, other than the identical operation, are distributed 
 
 in conjugate sets containing jp% ^^, operations, the total 
 
 number of operations oi G ial+p^ + p^ + ; and therefore 
 
 f^=\+p'- + p^ + 
 
 This equation can only be true if p'"' — 1 of the indices 
 a, ^ are zero, rj being some integer not less than unity. 
 
62 GENERAL [52 
 
 Hence G must contain p''' self-conjugate operations, which form 
 (I 27) a self-conjugate sub-group\ Hence : — 
 
 Theorem I. Every group whose order is the power of a 
 prime contains self-conjugate operations, other than the identical 
 operation ; and no such group can he simple. 
 
 53. Let Hi be the sub-group of G of order p^', formed of 
 
 G 
 
 its self-conjugate operations. Then ^ is a group of order 
 
 -"i 
 ^"""^i, and it must contain self-conjugate operations other 
 than identity, forming a self-conjugate sub-group of order 
 
 P 
 p''^. Corresponding to this self-conjugate sub-group o{ -= , Q 
 
 must have a self-conjugate sub-group H^ of order p^'+''', which 
 
 C 
 contains H^. If r^ + r^Km, jj is a group of order ^'"-'•■-'■j; and 
 
 -"2 
 
 it again will have self-conjugate operations forming a sub- 
 group of order p^'. Corresponding to this, G will have a self- 
 conjugate sub-group Hs of order ^'■i+''2+''«, which contains H,. 
 If the process thus indicated is continued, till G itself is arrived 
 at, a series of groups 
 
 Hi, H^, , Hn, G 
 
 is formed, each of which is self-conjugate in G and contains all 
 
 rr 
 
 that precede itl Moreover S^^ is the group formed of the 
 
 self-conjugate operations of -^ , and it is therefore an Abelian 
 
 group. 
 
 The series of sub-groups thus arrived at is of considerable 
 importance. From the mode of formation, it is clear that a 
 second group G' of order p^ cannot be simply isomorphic with 
 
 G, unless S^^ and -jtP are simply isomorphic for each value 
 
 > Sylow, Math. Ann. v. (1872), p. 588. 
 
 2 Let P be an operation or subgroup of G, and let if be a subgroup of G 
 which contains P. If every operation of H is permutable with P, we shall 
 speak of P as " self-conjugate in H." The phrase is to be regarded as 
 emphasizing the relation of P to a sub-group of G which contains it; and not 
 as implying that H is the greatest, or the only, subgroup of Q, within which P 
 is a self-conjugate operation or subgroup. 
 
53] PROPERTIES 63 
 
 of r. It by no means however follows that, when this latter 
 condition is satisfied, G and 0' are always simply isomor- 
 phic. This is indeed not necessarily the case; instances 
 to the contrary will be fonnd among the groups of order 
 p' and p* in §§ 69 — 72. While in general there is no limitation 
 
 on the type of any Abelian factor-group -^ formed from 
 
 two consecutive terms of the series, it is to be noted that 
 
 the last factor-group ^ cannot be cyclical \ a restriction that 
 
 C 
 
 can be established as follows. If -jjr- were cyclical, then, because 
 
 II n 
 
 TJ 
 
 jf^ is the group formed of the self-conjugate operations of 
 
 G- C 
 
 jf — , the operations of jf — could be arranged in the sets 
 
 -"« p -"» pa -"« p»«-i -"" 
 
 TT ' TT ^ TJ >'■•} -^ TT f 
 
 ■"»— I -"»— 1 -"m— 1 -On— I 
 
 C TT 
 
 p" being the order of -jj- and P^" being an operation of ^^ . 
 
 -"n -"n— 1 
 
 But since P is permutable with its own powers and with every 
 
 TT 
 
 operation of „ " , these operations would form an Abelian 
 
 G 
 group. Now the self-conjugate operations of jf — form the 
 
 •"n— 1 
 
 H C 
 
 group „ " , and therefore ^ — cannot be Abelian. Hence 
 
 ■"71—1 -tln^L 
 
 the assumption, that -jr- is cyclical, is incorrect. 
 
 -tin 
 
 One immediate consequence of this result is that a group G of 
 order/)'' is necessarily Abelian. For if it were not, its self-conjugate 
 
 operations would form a sub-group H of order p, and -^ being of 
 
 order p would be cyclical. 
 
 A second consequence is that if G, of order jo™, is not Abelian, 
 the order of the sub-group Hj^, formed of the self-conjugate operations 
 of G, cannot exceed p'^'". 
 
 ' Young, Amencan Journal, vol. xv (1893), p. 132. 
 
64 GENERAL [54 
 
 54. It was proved in the last chapter that an Abelian 
 group has one or more sub-groups the order of which is any 
 
 given factor of the order of the group itself. Hence, since ''+ ^ 
 
 Hf 
 is Abelian, i/^+i must have one or more sub-groups which contain 
 Hr, of any given order greater than that of H^ and less than its 
 own. 
 
 In particular, if p^ and p* are the orders of H,. and Hr+ 
 then -^^ must have a series of sub-groups of orders 
 
 p'-*-^p'-'-^ ,p, 1, 
 
 each of which contains those that follow it, while each one is of 
 
 necessity self-conjiigate in ^ . Hence between Hr and Hr+u a 
 
 series of groups h,+i, A^+j, , Aj_i of orders p'+\p'+'', p*-^ 
 
 can be chosen, so that each of the series 
 
 -"r> "s+i) "Jj+s, , A(_i, Ilr+i 
 
 is self-conjugate in G and contains all that precede it. We can 
 therefore always form, in at least one way, a series of sub-groups 
 of of orders 
 
 hp,p', ,p'^\p^, 
 
 such that each is self-conjugate in and contains all that 
 precede it. 
 
 It will be shewn in the next paragraph that every sub- 
 group of order p' of is contained self-conjugately in some 
 sub-group of order p'+\ and that every sub-group of order p^ of 
 a group of order ^J*"*"^ is self-conjugate. Assuming this result, it 
 follows at once that a series of sub-groups of orders 
 
 l,V'F'' p'^--\p^ 
 
 can always be formed, which shall contain any one given 
 sub-group and which shall be such that each group of the 
 series contains the previous one self-conjugately. It will not 
 however, in general, be the case, as in the previous series, that 
 each group of the series is self-conjugate in 0. 
 
 55. Let Gg be any sub-group of of order p*, and let Hr+i 
 be the first of the series of sub-groups 
 
 ■"1; -"2) ) -tin 
 
55] PEOPERTIES 65 
 
 which is not contained in Gs. Then -^ is a sub-group of ~ 
 which does not contain all the operations of "^tl. Every 
 
 SI f 
 
 operation of -~ is self-conjugate in ==- ; and therefore -fe is 
 
 ■"r Mr H,. 
 
 self-conjugate m ]g^ , "^[ > ^ group of order greater than its 
 
 own. Hence Og must be contained self-conjugately in some 
 
 group Og^t of order p«+*, where t is not less than unity. 
 ft 
 
 Moreover since -^^ must contain operations of order p, there 
 
 must be one or more groups of order p*+i which contain 0^ 
 self-conjugately. Hence : — 
 
 Theorem II. If Gg of order p^ is a sub-group of G, which 
 is of order p™-, then G must contain a sub-group of order p^+*, 
 t-^1, within which Gg is self-conjugate. In particular, every 
 sub-group of order p™-i of G is a self -conjugate sub-group ^ 
 
 Suppose now that Gs+t, of order p*+*, is the greatest sub- 
 group of G which contains a given sub-group Gg, of order p'^, 
 self-conjugately; so that Gg is one of |)™-«-* conjugate sub- 
 gi-oups. Suppose also that Gg+t+u, of order p«+t+u^ ^^ ^ j^^ jg 
 the greatest sub-group of G that contains Gg+t self-conjugately. 
 Every operation of Gg+t+u transforms Gg+t into itself; and no 
 operation of Gg^t+u that is not contained in Gg+t transforms Gg 
 into itself Hence, in Gg+t+u, Gg is one of p'^ conjugate sub- 
 groups and each of these is self-conjugate in (?«+;. 
 
 The p'"— »-* sub-groups conjugate with Gg may therefore be 
 divided into p'»-s-«-« sets of p" each, (u-^l), such that any 
 operation of one of the sets transforms each sub-group of that 
 set into itself 
 
 Similarly if Gr, of order p'', is the greatest sub-group of G 
 that contains a given operation P self-conjugately, and if Gr+g, 
 s -"^ 1, is the greatest sub-group that contains Gr self-conjugately, 
 then Gr must contain self-conjugately p^ operations of the 
 
 1 Frobenius, "Ueber endliche Gruppen," Berliner Sitzungsberichte (1895), 
 p. 173: Burnside, "Notes on the theory of groups of finite order," Proc. London 
 Mathematical Society, Vol. xxvi (1895), p. 209. 
 
 B. 5 
 
66 CONJUGATE [55 
 
 conjugate set to which P belongs, and therefore any two of 
 these ^* operations are permutable. Hence the p™^' conjugate 
 operations of the set to which P belongs can be divided into 
 j-^m-r-s ggtg Qf ps each, (s <t 1), such that all the operations of 
 any one set are permutable with each other. In particular, if 
 P is one of a set of p conjugate operations, all the operations of 
 the set are permutable. 
 
 56. If P is an operation of G, of order p", which is conju- 
 gate to one of its own powers P", there must be some other 
 operation Q such that 
 
 Q-'PQ = P-. 
 
 From this equation it follows that 
 
 Q-^P^Q = {Qr-'PQT = P-^°. 
 
 Again 
 
 Q-^P(^=Q-iP'^Q = pa^^ 
 
 Q-'P(^=Q-'P-'Q=P-', 
 
 and so on. Hence 
 
 Q-vp^Qv = pxo.'_ 
 
 If Q^ is the lowest power of Q which is permutable with P, 
 then in (P, Q} P will be one of /3 conjugate operations, and 
 therefore /3 must be a power oi p, say p^, less than p". Now 
 
 Q-P'PQP' = P-'''; 
 and therefore a^' = 1 (mod. p"), 
 
 while a3''"'^l (mod. p"). 
 
 First, we will suppose that p is an odd prime. Then since 
 xP' = X (mod. p), 
 whatever integer x may be, we may assume that a=l+kp', 
 where k is not a multiple of p ; and then 
 
 aP' = l +kp'-+'+ 
 
 aP"' = I + kp"-*'-' + 
 
 Hence r + s = n, 
 
 ■ and a = 1 (mod. p"-^). 
 
 In particular, we see that no operation of order p can be 
 conjugate to one of its powers. Hence if P and P' are two 
 
57] OPERATIONS 67 
 
 conjugate operations of order p, {P} and {P'] have no operation 
 in common except identity. Also, if {P} be a self-conjugate 
 sub-group of order p, each of its operations is self-conjugate. 
 
 If p is 2, we must take 
 
 a=± l+k2', 
 
 where k is odd, and we are led by the same process to the 
 result 
 
 a = + 1 (mod. 2"-'-). 
 
 57. If P is an operation of G which belongs to the sub- 
 group Hr+i (§ 53) and to no previous sub-group in the set 
 
 1, H^, H„_, , Hn, G, 
 
 then {P, H,] is a sub-group of Hj+j ; and therefore every 
 
 operation of ' — 'rf i^ self-conjugate in ^ . The set of 
 
 operations PHr is therefore transformed into itself by every 
 operation of G, and hence every operation that is conjugate to 
 P is contained in the set PHr- Suppose now, if possible, that 
 all the operations conjugate with P are contained in the set 
 PHr-i- Then every operation of O transforms this set into 
 
 f p TT 1 
 
 itself, and therefore every operation of ' '„ *"" is self-conjugate 
 
 in Yf — • ^^i^ however cannot be the case, since by supposition 
 
 P does not belong to H,- Hence among the operations 
 conjugate to P, there must be some which belong to PH^ 
 and not to PHr-i. In particular, if P is one of ^ conjugate 
 operations, no operation conjugate to P can belong to the set 
 PHy_^. For there must be an operation Q such that Q~^PQ 
 belongs to PHr and not to PHy-i ; and this being the case, no 
 
 one of the operations Qr^PQ', {x = l, 2, , ^ - 1 ), can belong 
 
 to PHr-i. But these operations with P constitute the conjugate 
 set. 
 
 It follows from the above reasoning that, if P belongs to 
 Hr+^, then each operation of the set P-'Q-^P'Q, where Q is any 
 operation of G, belongs to H,.. If Q belongs to Hs+, , (s < r), the 
 operation P-»Q-iP»Q, regarded as the product of P-^Q-^P" and 
 Q, must similarly belong to Hs. Hence unless every operation 
 
 5—2 
 
G8 SPECIAL TYPES [57 
 
 of Hg^y is permutable with P, there must be operations conju- 
 gate to P, which belong to the set PH^. Moreover those 
 operations of G which transform P into operations of the set 
 PHs form a sub-group. For if 
 
 8-'PS = PK, 
 
 and S'-'P8' = Ph', 
 
 then 8'-^8-'PS8' = PK ■ 8'-%S, 
 
 and when both hg and hj belong to Hg, so also does hs'S'~%S'. 
 
 58. In illustration of the foregoing paragraph, we will consider 
 a group G of order p''+* in which the group ff^ is of type (r), while 
 
 -=- is an Abelian group of type (1, 1, with « units). 
 
 Let Q be an operation of order p'' that generates H^ ; and let 
 
 Pi, Po, , Pghe s operations no one of which is a power of any 
 
 other, such that, with Q, they generate G. 
 
 If Pj and Pj are not permutable, then 
 
 and P^-vp^Pf = P^QP^. 
 
 Now Pj* belongs to H-^, and therefore 
 
 so that a = kp^'^. 
 
 Hence the only operations which are conjugate to Pi are 
 
 Pi, P,Q''-\ PiS^^'", , Pi^l^^^-'-^ 
 
 and Pi is one of a set of p conjugate operations, which are trans- 
 formed among themselves by Pj. Similarly Pj is one of a set of p 
 conjugate operations which are transformed among themselves by 
 Pi. Hence both Pi and P^ are permutable with each of the 
 
 operations Pg, P4, , P^. Again, Pg is not self-conjugate, and 
 
 there must therefore be another, say P^, of the set P3, P4, , P, 
 
 which, with its powers, transforms Pg into a set of p conjugate 
 operations. Then P3 will similarly transform P4; and P3, P4 will 
 
 be permutable with each of the set Pj, , Pj, Pi, P^. Hence 
 
 finally the set of s operations Pj, P^, , P, must be divisible 
 
 into sets of two, such that each pair are permutable with all the 
 remaining operations, but are not permutable with each other. The 
 group can therefore only exist if s is even^ 
 
 ' Young, " On groups whose order is the power of a prime," American 
 Journal of Mathematics, Vol. xv (1893), p. 171, 
 
59] OF GROUP 69 
 
 The sub-group {P,, P^, Q] is a group which satisfies the same 
 conditions as G, when s = 2. Its order is p""^, and it contains a 
 self-conjugate operation of order f. Now we shall see in §§ 65, 66 
 that such a group is necessarily of one of two types', the generating 
 operations of which satisfy one or the other of the sets of equations 
 
 ^^'=1, P/=l, P/=l, Ff^P,P, = P,QP'-': 
 
 or e*'=l, PP^Q, P/=l, Pf^P,P,= P,Qp'-\ 
 
 The same is true for the sub-groups {P^, P^, Q\, etc. ; and all the 
 operations of any one of these sub-groups are permutable with each 
 operation of {P^, P^, Q}. 
 
 Hence finally, since the equations to be satisfied by each pair of 
 operations, such as P^ and P^, may be chosen in two distinct ways, 
 
 there are in all 2 distinct types of group of order p'"^", for which 
 Hi is a cyclical group of order p'', and ^jr is an Abelian group of type 
 (1, 1, to s units). 
 
 59. It has been seen in § 55 that every sub-group G' of 
 
 order p^-^ of a group of order p™ is self-conjugate. Suppose 
 
 now that G contains two such sub-groups G' and G". Then 
 
 since G' and G" are permutable with each other, while the 
 
 order of {G', G"} is jp™, the order of the greatest group g' 
 
 common to them must (§ 33) be j)™"^; and since g' is the 
 
 greatest common sub-group of two self-conjugate sub-groups of 
 
 G, it must itself be a self-conjugate sub-group of G. The factor 
 
 G G' 
 
 group -; of order p' contains the two distinct sub-groups — ^ 
 
 if ^ 
 
 C" 
 
 and — r , which are of order p and permutable with each other. 
 ' Let S'i''i = l, S/2=1 , S>=1, 
 
 fl{Si) = l, /2(S«) = 1, fk(Si) = l, 
 
 Vfheiefj{Si) is an abbreviation for an expression of the form 8p"'Sj> 5/, be a 
 
 set of relations, such as was considered in § 18, which completely specify a 
 group. We may then, without altering the sense in which the word "type" 
 has been used (§§ 19, 44), speak of the type of group defined by these relations. 
 It is essential however that the relations should completely specify the group, 
 as otherwise they will define more than one type. For instance, it is clear, from 
 § 56, that the equations 
 
 QP = 1, PP' = 1, Q-iP(3=P", 
 where pis a, given prime, but a is not given, will define more than one type of 
 group. Any data in fact, which completely specify a group, may be said to 
 define a type of group. Thus in dealing with Abelian groups of order p'", a, type 
 is defined by a partition of m. 
 
70 NUMBER OF SUB-GROUPS [59 
 
 Hence — must be an Abelian group of type (1, 1), and 
 
 it therefore contains (§ 49) p + 1 sub-groups of order p. Hence 
 besides G', G must contain p other sub-groups of order p™-' 
 which have in common with G' the sub-group g'. If the « -(- 1 
 sub-groups thus obtained do not exhaust the sub-groups of Q 
 of order p™-\ let G'" be a new one. Then, as before, & and G'" 
 
 must have a common sub-group g", of order p'"'^, which is 
 
 fii/f 
 self- conjugate m. G. If g" were the same as g', —/- would be 
 
 G . ^ 
 
 contained in — , which by supposition is not the case. It may 
 
 now be shewn as above that there are, in addition to G', p 
 sub-groups of order p™-i which have in common with G' the 
 group g". These are therefore necessarily distinct from those 
 before obtained. This process may clearly be repeated till all 
 the sub-groups of order ^'"^^ are exhausted. Hence finally, if 
 the number of sub-groups of G, of order p™~i be r^-i, we have 
 rm-i = 1 (mod. p). 
 
 60. The self-conjugate operations of a gi-oup G of order 
 p^\ whose orders are p, form with identity a self-conjugate 
 sub-group whose order is some power of p ; and therefore their 
 number must be congruent to — 1, mod. p. On the other hand, 
 if P is any operation of G of order p which is not self-conjugate, 
 the number of operations in the conjugate set to which P 
 belongs is a power of p. Hence the total number of operations 
 of G, of order p, is congruent to — 1, mod. p. Now if r-^ is the 
 total number of sub-groups of G of order p, the number of 
 operations of order p is r^ip- 1), since no two of these sub- 
 groups can have a common operation, except identity. It 
 follows that 
 
 n (p - .1) s - 1, (mod. p), 
 
 and therefore r^ = 1, (mod. p). 
 
 If now Gg is any sub-group of G of order p\ and if Gs+t is the 
 greatest sub-group of G in which Gs is contained self-conjugately, 
 then every sub-group of G which contains Gg self-conjugately is 
 contained in Gg+f But every sub-group of order p«+^ which 
 contains G^, contains Gs self-conjugately; and therefore every 
 
62] OF GIVEN ORDER 71 
 
 sub-group of order p^+\ which contains Qg, is itself contained in 
 Gs+t- By the preceding result, the number of sub-groups of 
 
 -^ of order p is congruent to unity, mod. p. Hence the 
 
 number of sub-groups of G of order p'+\ which contain Gs 
 of order p^, is congruent to unity, mod. p. 
 
 61. Let now r, represent the total number of sub-groups of 
 order p' contained in a group G of order p'"-. If any one of 
 them is contained in a^ sub-groups of order p*+\ and if any one 
 of the sub-groups of order p*+' contains by sub-groups of order 
 p'; then 
 
 2 a^= 2 by; 
 
 for the numbers on either side of this equation are both equal 
 to the number of sub-groups of order p''+^, when each of the 
 latter is reckoned once for every sub-group of order p^ that it 
 contains. It has however been shewn, in the two preceding 
 paragraphs, that for all values of x and y 
 
 tta; s 1, by=l (mod. p). 
 
 Hence rs = rs+i (mod. p). 
 
 Now it has just been proved that 
 
 ri = 1 and r^-i = 1 (mod. p) ; 
 
 and therefore finally, for all values of s, 
 
 rg = 1 (mod. p). 
 
 We may state the results thus obtained as follows : — 
 
 Theorem III. The number of sub-groups of any given 
 order p' of a group of order p"^ is congruent to tmity, mod. p *. 
 
 Corollary. The number of self-conjugate sub-groups of 
 order p^ of a group of order p™ is congruent to unity, mod p. 
 
 This is an immediate consequence of the theorem, since the 
 number of sub-groups in any conjugate set is a power of p. 
 
 62. Having shewn that the number of sub-groups of G of 
 order p' is of the form 1 + kp, we may now discuss under what 
 
 * Frobenius, " Verallgemeinerung des Sylow'schen Satzes," Berliner Sitz- 
 ungsberichte, (1895), p. 939. 
 
72 GROUPS WITH A SINGLE SUB-GROUP [62 
 
 circumstances it is possible for k to be zero, so that G then 
 contains only one sub-group G^ of order p'. 
 
 If this is the case, and if P is any operation of G not con- 
 tained in Gg, the order of P must be not less than 2^*+'; for if 
 it were less, G would have some sub-group of order p* containing 
 P and this would necessarily be different from Gg. If the order 
 of P is p*+*, then {P^'} is a cyclical sub-group of order pf ; and it 
 must coincide with Gg. Hence, if Gg is the only sub-group of 
 order p^, it must be cyclical. 
 
 Suppose now that G contains operations of order p^ (r >s), 
 but no operations of order p^+^ ; and let P be an operation of Q 
 of order p'^. Then [P] must be contained self-conjugately in a 
 non-cyclical sub-group of order p''+'. 
 
 We will take first the case in which p is an odd prime. 
 Then (§ 55) G must contain an operation P' which does not 
 belong to [P], such that 
 
 P'-^PP' = P"^^ P'p = P^ 
 
 If a were unity, [P, P'} would be an Abelian group of order 
 p''+^ containing no operation of order p''+^. Its type would 
 therefore be (r, 1), and it would necessarily contain an operation 
 of order p not occurring in {P}. It has been shewn that this is 
 impossible if Gs is the only sub-group of order p\ and therefore 
 a cannot be unity. 
 
 We may then without loss of generality (§ 56) assume that 
 a = 1 -F^'-i. 
 
 Moreover if /Q were not divisible by p, the order of P would 
 be p^+^, contrary to supposition. Hence we must have the 
 relations 
 
 p-^pp = pi+/~' p'p = pyp. 
 
 By successive applications of the first of these equations, we 
 get 
 
 pi-ypxp'v _ px(i-¥yp"'^) ^ 
 
 for all values of cc and y ; and from this it immediately follows 
 that 
 
 (pxpy — pppx{p+hp(p+'^)p''^) 
 
63] OF GIVEN ORDER 73 
 
 Hence if 
 
 ^ = F~'-7> 
 
 the order of P'^F, an operation not contained in (P), is p. This 
 is impossible if 0^ is the only sub-group of order p\ If then 
 r<m,,G must contain operations of order greater than p'' ; and 
 is therefore a cyclical group. Hence : — 
 
 Theorem IV. If 0, of order p"\ where p is an odd prime, 
 contains only one sub-group of order p\ G must he cyclical. 
 
 63. When p = 2, the result is not so simple. 
 
 Let Q be an operation that transforms {P\ into itself, and 
 suppose that the lowest power of Q that occurs in {P] is Q*. 
 Then [P, Qj must be defined by 
 
 q-'PQ = P% Q' = P^ 
 
 where a = ±l + k2'-^. 
 
 Moreover yS must be a multiple of 4, as otherwise the order of 
 Q would be greater than 2''- A simple calculation now gives 
 
 {P^Q^y = p/3+a;(l+a2) . 
 
 and since /3 = (mod. 4), 
 
 and a^ s 1 (mod. 2'-i), 
 
 X can always be chosen so that 
 
 ^ + oc(l+a!') = (mod. 2'). 
 
 When X is thus chosen, P^Q^ is an operation of order 2 which is 
 not contained in {P}. But this is inconsistent with Gg being 
 the only sub-group of order 2* ; hence [P] must contain Q'. 
 
 If now [P] is not a self-conjugate sub-group, there must 
 (§ 55) be some operation P' of order p'', conjugate with P and 
 not contained in [P], which transforms {P} into itself; and 
 then {P, P'} is defined by 
 
 P'-^PP' = P-, P'' = p^, 
 
 where a is — 1 or + 1 + 2''-^ and ^ is odd. 
 
 If a = 1 -f- 2'-\ 
 
 then (P^Py. = P^ {3+«(i+2'"') } ; 
 
 and X can be chosen so that P^P' is of order 2. Hence this 
 case cannot occur. 
 
74 GROUPS WITH A SINGLE SUB-GROUP [63 
 
 If a is either — 1 or - 1 + 2'-', the defining relations of 
 [P, P'} are not self-consistent unless r is 2 ; for they lead to 
 
 jp'—ip^^p' _ p-2^ 
 
 and P'-ip2?P' = p2^. 
 
 If r is 2, and if P" is an operation that transforms P into P', 
 
 so that 
 
 p"-ipp'' _ p' 
 
 then (PP'J = PP"2P' = P-^P', 
 
 since p2^p'2=p"2 
 
 Hence PP" would be of order 8 ; and therefore this case 
 cannot occur. 
 
 Hence, finally, \P] must be self-conjugate. 
 
 If r -I- 1 < m, {P] must be transformed into itself by some 
 operation Q' not contained in [P, Q], so that 
 
 Q-'PQ = p-i+*2"', 
 
 and Q'-'PQ' = P-i+*'2'"\ 
 
 where k and k' are each either zero or unity. If both are zero 
 or both unity, QQ' is permutable with P ; and if one is zero and 
 the other unity. 
 
 In either case, the group contains an, operation of order 2 that 
 does not belong to (Pj, and this is inadmissible. Hence, lastly, 
 r must not be less than m — 1. 
 
 Suppose now that r is equal to m — 1, and that of the 
 operations of G, not contained in {Pj, Q has as small an order 
 as possible, say 2^{t^s+l). Then Q^ of order 2«-i is contained 
 in {P] ; without loss of generality we may assume 
 
 Q= = P^"-'. 
 Moreover Q-'PQ = P'' or P-'+'""\ 
 
 and hence Qt^P^'^'Q = P-^"'"'. 
 
 Now Q is permutable with P^"", one of its own powers, and 
 therefore P^""' is an operation of order 2. Hence < is 2 and s 
 is unity. If then s is greater than unity, G must contain 
 
65] OF GIVEN ORDER 75 
 
 operations of order 2"' and must therefore be cyclical. Moreover 
 if s is unity, the relations 
 
 lead to 
 
 and they are therefore inadmissible. 
 
 On the other hand, the relations 
 
 Q--PQ = p-\ Q^ = P-^"'-\ 
 
 give {P'^Qf = P^""' ; 
 
 and every operation of G, not contained in [P\, is of order 4. 
 Also these relations are clearly self-consistent, and they define 
 a group of order 2™. Hence finally : — 
 
 Theorem Y. If a group G, of order 2"\ has a single 
 sub-group of order 2*, (s > 1), it must be cyclical ; if it has a 
 single sub-group of order 2, it is either cyclical or of the type 
 defined by 
 
 P='"-' = l, Q2 = P="'"', Q-'PQ = P-\ 
 
 64. We shall now proceed to discuss, in application of the 
 foregoing theorems and for the importance of the results them- 
 selves, the various types of groups of order p™ which contain 
 self-conjugate cyclical sub-groups of orders p'''-^ and p'""" 
 respectively. It is clear from Theorem V that the case p = 2 
 requires independent investigation ; we shall only deal in detail 
 with the case in which p is an odd prime, and shall state the 
 results for the case when p = 2. 
 
 The types of Abelian groups of order ^"' which contain 
 operations of order p™-^ are those corresponding to the symbols 
 (m), (m— 1, 1), (m — 2, 2), and (m — 2, 1, 1). We will assume 
 that the groups which we consider in the following paragraphs 
 are not Abelian. 
 
 65. We will first consider a group G, of order p'"-, which 
 contains an operation P of order p™"'. The cyclical sub-group 
 {P] is self-conjugate and contains a single sub-group {P^" | of 
 order p. By Theorem IV, since G is not cyclical, it must 
 contain aa operation Q', of order p, which does not occur in [P]. 
 
76 SPECIAL TYPES [65 
 
 Since [P\ is self-conjugate and the group is not Abelian, Q 
 must transform P into one of its own powers. Hence 
 
 Q'-^PQ' = P% 
 and since Q'p is permutable with P it follows, from § 56, that 
 
 a = 1 + %)'"-=. 
 
 Since the group is not Abelian, k cannot be zero ; but it may 
 have any value from 1 to _p — 1. If now 
 
 kx=l (mod. p), 
 
 then Q'-:»PQ'=" = P^+p'"''; 
 
 and therefore, writing Q for Q''^, the group is defined by 
 
 pp"-' = 1 , QP=l, Q-^PQ = P^+P-'. 
 
 These relations are clearly self-consistent, and they define a 
 group of order p^. 
 
 There is therefore a single type of non-Abelian group of 
 order p™ which contains operations of order j9™~S because, for 
 any such group, a pair of generating operations may be chosen 
 which satisfy the above relations. 
 
 From the relation 
 
 Q-'PQ = P'+i''"\ 
 it follows by repetition and multiplication that 
 
 Q-ypxQy = pam+yp"^'')^ 
 and therefore that 
 
 /pxQyy — pxz{i+i{z+i)yp''~''}{i-yzp'"^)Qy'^ 
 
 and {P^'Q'y = P^^. 
 
 Hence G contains p cyclical sub-groups of order p™-\ of 
 
 which P and PQ" (y=i,% p-1) may be taken as the 
 
 generating operations. Since Q and P* are permutable, G 
 also contains an Abelian non-cyclical sub-group [Q, P^] of 
 order p'"-^ It is easy to verify that the 1 4 jp sub-groups thus 
 obtained exhaust the sub-groups of order p'"-^ ; and that, for any 
 other order p% there are also exactly p + 1 sub-groups of which 
 p are cyclical and one is Abelian of type (s - 1, 1). 
 
 The reader will find it an instructive exercise to verify the 
 results of the corresponding case where ^ is 2 ; they may be stated 
 thus. There are four distinct types of non-Abelian group of order 
 
66] OF GROUP 77 
 
 2™, which c 
 
 one is the 
 
 defined by 
 
 2™, which contain operations of order 2™-^, when m>3. Of these, 
 one is the type given in Theorem V, and the remaining three are 
 
 P2""'=l, 
 
 Q'=l, QPQ = P'^''- 
 
 P'-'=1, 
 
 Q'=l, QPQ = P-^^'' 
 
 P"-'=1, 
 
 Q'=l, QPQ = p-\ 
 
 When m = 3, there are only two distinct types. In this case, the 
 second and the fourth of the above groups are identical, and the 
 third is Abelian. 
 
 66. Suppose next that G, a group of order p™, has a 
 self-conjugate cyclical sub-group {P} of order p'"^~'', and that 
 no operation of Q is of higher order than ^'"~'-. We may 
 at once distinguish two cases for separate discussion ; viz. 
 (i) that in which P is a self-conjugate operation, (ii) that in 
 which P is not self-conjugate. 
 
 Taking the first case, there can be no operation Q' in such 
 that Q'P' is the lowest power of Q' contained in [P] , for if there 
 were, {Q', P] would be Abelian and, its order being p'"^, it would 
 necessarily coincide with G. Hence any operation Q', not 
 contained in {P}, generates with P an Abelian group of type 
 (m — 2, 1), and we may choose P and Q as independent 
 generators of this sub-group, the order of Q being p. If now 
 R' is any operation of (r not contained in {Q, P], R'p must 
 occur in this sub-group, and therefore 
 
 If /3 were not zero, Rp' would be the lowest power of R 
 that occurs in [P], and we have just seen that this cannot be 
 the case. Hence 
 
 R'p = P«, 
 
 and {R', P} is again an Abelian group of type (m — 2, 1). If 
 P and R are independent generators of this group, the latter 
 cannot occur in {Q, P}. Now since Q is not self-conjugate, 
 
 R-'QR = QP^; 
 and since RP is permutable with Q 
 
 PP^=1, 
 so that /S = (mod. p"'-')- 
 
 Hence R-'QR = QP*^""', 
 
78 SPECIAL TYPES [6g 
 
 where k is not a multiple of p. If finally, P* be taken as a 
 generating operation in the place of P, the group is defined by 
 
 pj'"'-" = l, QP = \, BP = 1, R-'QR = QPP'"\ 
 
 PQ=QP, PR = RP. 
 
 There is therefore a single type of group of order p^, which 
 contains a self-conjugate operation of order p™— 2 a,nd no opera- 
 tion of order j)™~^. 
 
 67. Taking next the second case, we suppose that 
 contains a cyclical self-conjugate sub-group [P] of order p™~^, 
 but no self- conjugate operation of this order. 
 
 If G contains an operation Q' such that Q'p' is the lowest 
 power of Q' occurring in {P), it follows (§ 56), since {P} is 
 self-conjugate, that 
 
 Q'->PQ' = Pi+*i'""'. 
 
 Moreover, since the order of Q' cannot exceed p'^~^, we have 
 
 A simple calculation, similar to that of § 65, will now shew 
 that 
 
 (P'^Q'y = p («+«');>'' 
 
 while {P''Q'y = Q'^P^ {p+i^w+Dp""'}. 
 
 Hence P~°-Q' is an operation of order p'^, none of whose 
 powers, except identity, is contained in jP}. If this operation 
 be represented by Q, is defined by 
 
 PP-' = 1, Qi" = 1, Q-ipQ = pi+ip"-'. 
 If Z; is a multiple of p^ the group is Abelian. 
 If k is k'p, and if Q" is taken for a new generating operation, 
 where a is chosen so that 
 
 ak' =1 (mod. p), 
 we have a single type defined by 
 
 pp"-' =1, Qp' = 1, Q-^PQ = Pi+P"-'. 
 
 If A; is not a multiple of ^, we may take it equal to h + hp, 
 where k^ and k^ are both less than p, and A^i is not zero. (This 
 case cannot occur if m < 5.) Then 
 
 Q-x-x.ippQXi+x^ _ pi+h^xy'*+ (i,a;2+A:ia;,)p"'"'_ 
 
67] OF GROUP 79 
 
 when m > 5 ; while if m = 5, the coefficient of p"^-^, viz. of p-, in 
 the index of P must be increased by ^x^ («i — 1) A^i^. 
 
 If now we choose z^ so that 
 
 kjXj =1+ y,p, 
 and then X2 so that 
 
 yi + kiX2 + ksXi = (mod. p), 
 (with the suitable modification when m = 5), then Q^i+^^p 
 transforms P into P^+^ • We therefore again in this case 
 get a single type of group, which, if Q^^+^^^p is represented by 
 R, is defined by 
 
 Pp""' = 1 , Rp' = l, R-'PR = P^+p"'*. 
 
 As already stated, this type exists only when to > 4. 
 
 If every operation of G is such that its pth power occurs in 
 {P}, P must be one of a set of p conjugate operations ; for if P 
 is one of p^ conjugate operations, there must be an operation Q 
 which transforms P into pi+V ^ where k is not a multiple of p, 
 and then Qp' is the lowest power of Q which is permutable 
 with P Since P is one of p conjugate operations, it must be 
 self-conjugate in an Abelian group [Q, P} of type (m — 2, 1). 
 
 Again, since P andP^+* are conjugate operations, P must be 
 contained in a group of order j^3'"~^ defined by 
 
 PP"'' =1, PP = 1, R-'PR = P'+P"''. 
 
 If Q is not a self-conjugate operation, it must be one of p 
 conjugate operations and (§ 57) these must be 
 
 Q, QP^"", QP"^^", , QP'^^'p"". 
 
 Now if R-'QR = QP"*"""', 
 
 then R-'P--QR = P-^Q. 
 
 Hence, if Q is not self-conjugate, P~'^Q is a self-conjugate 
 operation of order /»™~°: and the group is that determined 
 in the last paragraph. Therefore Q must be self-conjugate, 
 and the group is defined by 
 
 pj''"-'=l, QP = \, RP=l, R-'PR = P'+P"'\ 
 
 PQ = QP, RQ = QR 
 
80 SPECIAL TYPES 
 
 [67 
 
 It is clear from these relations that the group thus arrived 
 at is the direct product of the groups {P, R] and {Q}.* 
 
 It is to be expected, from the result of the corresponding case at 
 the end of § 65, that the number of distinct types when » = 2 is 
 much greater than when p is an odd prime. There are, in fact 
 when m>5, fourteen distinct types of groups of order 2", which 
 contain a self-conjugate cyclical sub-group of order 2"-^ and no 
 operation of order 2™-i They may be classified as follows. 
 
 Suppose first that the group has a self-conjugate operation of 
 order 2""-^ There is then a single type defined by the relations 
 
 (i) A^' =1, B'=l, C'=l, CBC = BA 
 
 ,m— 8 
 
 Suppose next that the group G has no self-conjugate operation 
 of order 2"-^, and let [A] be a self -conjugate cyclical sub-group of 
 
 order 2"-^. If — r is cyclical, there are, when «i > 5, five distinct 
 
 types. The common defining relations of these are 
 
 ^=""'=1, 5^=1, B--'AB = A°-; 
 
 and the five distinct types are 
 
 (ii) a = - 1, (iii) a = 1 -l- 2™"', (iv) a=-\+ 2""-^ 
 
 (v) a=l-l-2"-^ (vi) a=-l + 2" 
 
 am-4 
 
 * In each of tbe five distinct eases to which we have been led in the discussion 
 contained in §§ 65-67, we have arrived at a set of defining relations, containing 
 no indeterminate symbols, such that in each ease a set of generating operations 
 can be chosen to satisfy these relations. To justify the statement, in each 
 particular ease, that such a set of relations gives a distinct type of group, it is 
 finally necessary to verify that the relations actually define a group of order p". 
 In the cases dealt with in the text, this verification is implicitly contained in the 
 process by which the relations have been arrived at. We have therefore omitted 
 the direct verification, which moreover is extremely simple. We shall similarly 
 omit the corresponding verifications in the discussion of groups of orders p* and 
 •p*, as in none of these cases does it present any difficulty. 
 
 To illustrate the necessity of such a verification in general, we may consider 
 a simple case. The relations 
 
 P3 = l, ©9=1, P-1QP=(3«, 
 
 where a is any given integer, certainly define a group whose order is equal to or 
 is a factor of 27, since they indicate that {P} and {Q} are permutable. They will 
 not however give a type of group of order 27, imless a is 1, 4 or 7. For instance, 
 if o = 5, the relations involve 
 
 Q=P-'QP^=Q\ or Q'=l. 
 Hence Q=Q*--!-^-^=l, 
 
 and the relations hold only for a group of order 3. 
 Again, if a=3, the relations give 
 
 p-ig3p=Q3 = i, or e' = l, 
 ind as before they define a group of order 3. 
 
68] OF GROUP 81 
 
 If m = 5, then (iv) and (v) are identical, and (vi) is Abelian ; so that 
 there are only three distinct types. If jw = 4, there is a single type ; 
 it is given by (ii). 
 
 G 
 
 When j-jT is not cyclical, the square of every operation of G is 
 
 contained in \A\. If all the self-conjugate operations of G are not 
 contained in \A\, there must be a self- conjugate operation 5, of order 
 2, which does not occur in {A\. If C is any operation of G, not 
 contained in \A, E\, then {A, G\ is a self-conjugate sub-group of 
 order 2™~', which has no operation except identity in common with 
 \B\. Hence (? is a direct product of a group of order 2 and a group 
 of order 2™"''. There are therefore, for this case, four tj'pes (vii), 
 (viii), (ix), (x), when ?n > 4, corresponding to the four groups of 
 order 2™"^ of § 65. If m = 4, there are two types. 
 
 Next, let all the self -conjugate operations of G be contained in 
 \A\ \ and suppose that A is one of two conjugate operations. Then 
 G must contain an Abelian sub-group of type (m — 2, 1), in which A 
 occurs ; and it may be shewn that, when m > 4, there are two types 
 defined by the relations 
 
 A^''=\, £^=l, BAB = A, 0^=1, 
 (xi) and (xii) ^^^^^^,™-s ^^^^^_, ^^ ^_,,,.-»^ 
 
 When 7n = 4, there is, for this case, no type. 
 
 Lastly, suppose that A is one of four conjugate operations. Then 
 G must contain sub-groups of order 2'"-', of the second and the 
 third types of § 65, and a sub-group of order 2™"^ of either the 
 first or fourth type (l. c). In this last case, there are two distinct 
 types defined by 
 
 A^''~'=l, 5^=1, JBAB = A^+''° . 
 (xiii) and (xhr) „ ., ™_3 
 
 ^ ' ^ 'C'=l, CAC^A--"*' , CBC = B or BA'. 
 
 These two types exist only when m>i. 
 
 68. We shall now, as a final illustration, determine and 
 tabulate all types of groups of orders p% p^ and p^. It has been 
 already seen that when p = 2 the discussion must, in part at least, 
 be distinct from that for an odd prime; for the sake of brevity 
 we shall not deal in detail with this case, but shall state the results 
 only and leave their verification as an exercise to the reader. 
 
 It has been shewn (§ 53) that all groups of order p^ are Abelian ; 
 and hence the only distinct types are those represented by (2) and 
 (1, !)• 
 
 For Abelian groups of order p% the distinct types are (3), (2, 1) 
 and (1, 1, 1). 
 
 B. 6 
 
 .nra-s 
 
82 SPECIAL TYPES [68 
 
 If a non-Abelian group of order j? contains an operation of 
 order p^, the sub-group it generates is self -conjugate ; hence (§ 65) 
 in this case there is a single type of group defined by 
 
 If there is no operation of order p', then since there must be a 
 self-conjugate operation of order p, the group comes under the head 
 discussed in § 66 ; there is again a single type of group defined by 
 
 PP = 1, (p=\, Bo^l, R-'QR = QP, 
 
 R-^PR = P, Q-'PQ = P. 
 
 These two types exhaust all the possibilities for non-Abelian 
 groups of order p'. 
 
 69. For Abelian groups of order p\ the possible distinct types 
 are (4), (3, 1), (2, 2), (2, 1, 1) and (1, 1, 1, 1). 
 
 For non-Abelian groups of order p'^ which contain operations of 
 order p^ there is a single type, namely that given in § 65 when m 
 is put equal to 4. 
 
 For non-Abelian groups which contain a self-conjugate cyclical 
 sub-group of order p'' and no operation of order p^, there are three 
 distinct types, obtained by writing 4 for m in the group of § 66 and 
 in the first and the last groups of § 67. The defining relations of 
 these need not be here repeated, as they will be given in the sum- 
 marizing table (§ 73). 
 
 It remains now to determine all distinct types of groups of 
 order p'^, which contain no operation of order p^ and no self-conjugate 
 cyclical sub-group of order p^. We shall first deal with groups 
 which contain operations of order p^. 
 
 Let S be an operation of order p^ in a group G of order p*. The 
 cyclical sub-group {S} must be self-conjugate in a non-cyclical sub- 
 group {S, T} of order p^, defined by 
 
 If R is any operation of G, not contained in {S, T], then since 
 {>S^} is not self -conjugate, we must have (§ 57) 
 
 and therefore R-'S^ R = S^. 
 
 Now T-'S'T-^^, 
 
 and therefore the pth power of every operation of order p^ in ff is a 
 self -conjugate operation. 
 
 First let us suppose that G contains other self-conjugate 
 operations besides those of {S']. Every such operation must occur 
 
70] OF GROUP 83 
 
 in the group that contains {.S^} self-conjugately ; hence in this case 
 T must be self-conjugate. 
 
 We now therefore have 
 
 R-^SR = 5'+"^r^ H-'TR = T, 
 These equations give 
 
 Hence, if 8 = 0, S~'''R is an operation of order p. Denoting this 
 by Ji' and S'-'T^ by T', the group is defined by 
 
 Si"=l, T'f^l, R'P^l, R'-\SR' = ST', 
 
 r-^ST' = S, R'-^T'R' = T'. 
 
 5a 
 
 On the other hand, if 8 is not zero, S^ ^ R is an operation of 
 order p' such that R transforms it into a power of itself. This is 
 contrary to the supposition that the group contains no cyclical self- 
 conjugate sub-group of order p^. Hence S cannot be different from 
 zero : we therefore have only one type of group. 
 
 70. Next, let [S^] contain all the self-conjugate operations of G; 
 and as before, let {S, T\ be the group that contains {S] self- 
 conjugately. If G contains an operation S' of order p^ which 
 does not occur in [S, T}, there must also be a non-cyclical sub- 
 group {<S", T'\ of order p^ which contains {S'\ self-conjugately. 
 Now [S, T] and {iS", T'} must have a common sub-group of order p- ; 
 since this is self- conjugate in G, it cannot be cyclical. The only 
 non-cyclical sub-groups of orders p'^ that {S, T] and {<S", T'\ contain 
 are {(S*", T} and {S"'', 'f'}. Hence these must be identical, and 
 therefore T must occur in {S', T'}. If now {S, T) and {S', T'] were 
 both Abelian, T would be permutable both with S and with aS", and 
 would therefore, contrary to supposition, be a self -conjugate 
 operation. Hence either (i) G must contain a non- Abelian group 
 {S, T\ of order p'^, in which S is an operation of order p'^ ; or (ii) the 
 Abelian group {*S', T], in which S is an operation of order p^, must 
 contain all the operations of G of order p'. 
 
 In the case (i), the group is defined by 
 
 SP^=l, TP=l, T-'ST = S'*^, 
 
 R~^SR = S'-"'i'T^, R-^TR = SyPT, 
 
 R'> = S^P. 
 
 6—2 
 
84 SPECIAL TYPES [70 
 
 These equations give 
 
 and therefore, except when p = 3, S'^B is an operation of order p. 
 Hence, if p > 3, we may, without loss of generality, put zero for 8, 
 In this case a simple calculation gives 
 
 where u^ = |/3ya;(a:- 1) + a;a, P^ = x^, y.^ = xy. 
 
 Hence if B^ = R', 8^ = 8', S''PT=r, where x, y and z are not 
 multiples of p, then 
 
 S'p^ =1, T'P=\, T'-hS'T' = ^'1+^, 
 
 Ji'-^S'R' = S''+K-^^.)i>T'vP , 
 
 y'P 
 R'-TR' = S' V T', R'P=l. 
 If now we take 
 
 then R'-^S'R' = S'T', R'-'T'R' = S"^Py'>T'. 
 
 Dropping the accents, the defining relations now become 
 
 R-'SR = ST, R-^TR = S''^T; 
 
 where a is either zero, unity, or any given non-residue. Since the 
 sub-group {R, T, S'^] contains all the operations of order p of the 
 group, it follows at once that these three cases give three distinct 
 types. 
 
 When jB = 3, it will be found that the defining relations again 
 give three distinct types. The operation R may always be chosen so 
 that a and y are zero. If it is so chosen, the three types correspond 
 to the values 1, ; — 1, 1 ; and — 1, — 1 ; of ^8 and 8. 
 
 In case (ii), the group is defined by 
 
 R-'^SR^S^+'^'TP, R-^TR = Syi'T, 
 
 RP^l; 
 
 with the condition that all operations of G, not contained in {S, T}, 
 are of order p. 
 
 The formulifi for R-^'SR' and R-'TR^ enable us to calculate 
 directly the power of any given operation of G. Thus they give 
 
 {S''R)P = ,SJ'^{i+i^r(p+i)p(p-i)+i«i'(p+i)} . 
 
72] OF GROUP 85 
 
 If p>3, this gives 
 
 so that, if X is not a multiple of p, the order of S^E is p^. Hence 
 the type of group under consideration can only occur when p = 3. 
 
 In this case {S'Bf = S^(.^+Py) . 
 
 Hence if p = 3 and (iy = -l (mod. 3), we obtain a new type. A 
 reduction similar to that in the previous case may now be effected ; 
 and taking unity for x, the group is defined by 
 
 S^=l, T^=l, £^=1, T-'ST:=S, 
 
 71. It only remains to determine the distinct types which 
 contain no operation of order p^. 
 
 Suppose first that the self -conjugate operations of G form a 
 group of order p^. This must be generated by two independent 
 operations P and Q of order p. 
 
 If now R is any other operation of the group, {P, Q, R] must be 
 an Abelian group of type (1, 1, 1). If again S is any operation not 
 contained in {P, Q, R\, it cannot be permutable with R ; for if it 
 were, R would be self -conjugate. There must therefore be a relation 
 of the form (§57) 
 
 S-'RS = RP<^Q^ 
 
 Since any operation of {P, Q] may be taken for one of its 
 generating operations, we may take P'^Q^ or P for one. If then Q' 
 is an independent operation of {P, Q], G will be defined by 
 
 FP=1, G'^ = l, ^"=1, ^^ = 1, S-''RS = RP', 
 
 in addition to the relations expressing that P' and Q' are self- 
 conjugate. There is then in this case a single type. That all the 
 operations of G in this case are actually of order p follows from the 
 obvious fact that every sub-group of order 2^ is either Abelian or of 
 the second type of non- Abelian groups of order p^. 
 
 72. Suppose, secondly, that the self-conjugate operations of G 
 form a sub-group of order p, generated by P. There must then be 
 some operation Q which belongs to a set of p conjugate operations ; 
 for if every operation of G which is not self-conjugate were one of 
 a set of p^ conjugate operations, the total number of operations in 
 the group would be congruent to p (mod. p^). It follows that Q 
 must be self-conjugate in a group of order p" ; and since P is also 
 self-conjugate in this group, it must be Abelian. Let P, Q, R be 
 generators of this group and ;S' any operation of G not contained in 
 
86 SPECIAL TYPES OF GROUP [72 
 
 it. We may now assume that Q belongs to the sub-group H 
 (§ 53), and therefore that 
 
 while S-'IiS=IiQ^py. 
 
 1 _i 
 If j3 were zero, Q'^S y would be a self-conjugate operation not 
 contained in {P} ; and therefore j8 must be different from zero. We 
 may now put 
 
 Q^py = Q', P-P = P'; 
 
 and the group is then defined by the relations 
 
 P'3'=l, ^'^=1, B^ = l, 8" = !, S-'BS = EQ', S-'Q'S=Q'F, 
 
 together with the relations expressing that P is self-conjugate. 
 There is thus again in this case, at most, a single type. It remains 
 to determine whether the operations are all actually of order }). 
 
 The defining relations give 
 
 S-'^P'^'Q^'R^'S = P'''+^Q^'+^Ry'+^, 
 where (^x+i = "-=! + ^x, ^x+i = Px + yx, ya:+i = yx; 
 
 and therefore S'^'P'^Q^ R^ S'' ^P^'Q^'R^', 
 
 where Oi^^a + x^ + }^x{x-l)y, p^ = ^ + xy, yj; = >'. 
 
 Hence 
 
 {P'^Q^RySy^^P^'^Q^^'R^^' 
 
 If /) is a greater prime than 3, the indices of P, Q, R are all 
 multiples of p; hence P'^Q^RyS is of order p, S being any opera- 
 tion not contained in G. If however p = B, then 
 
 {P''Q^RyS)' = Py, 
 
 so that, if y is not a multiple of 3, P'Q^RyS is an operation of 
 order 9. Hence this last type of group exists as a distinct type for 
 all primes greater than 3 ; but for 2^ = 3, it is not distinct from one 
 of the previous types containing operations of order 9. 
 
 73. In tabulating, as follows, the types of group thus 
 obtained, we give with each group G a symbol of the form 
 (a,b, ){a',b', )(a",b", ) 
 
 indicating the types of H,^, -jr , jr ' , where 
 
 -"1 -"2 
 
 Hi, H2, Hg, , Hn, Gr 
 
 is the series of self-conjugate sub-groups defined in §53. This 
 
73] TABLES 87 
 
 symbol is to be read from the left so that (a, b, ) is the 
 
 type of Hi. 
 
 Moreover in each group there is no operation of higher 
 order than that denoted by P. 
 
 Table of groups of order p", p an odd prime*. 
 
 I. n = 2, two types, 
 (i) (2); (ii) (1,1). 
 
 II. « = 3, five types. 
 
 (i) (3); (ii) (2,1); (iii) (1,1,1); 
 (iv) PP' = 1,QP = 1, Q-'PQ = P^+P, (1) (11) ; 
 (v) PP = 1,QP=1,PJ!>=1, R-^QR = QP, 
 R-'PR = P, Q-'PQ = P, (1) (11). 
 
 III. n = 4, fifteen types. 
 
 (i) (4); (ii) (3, 1); (iii) (2, 2); (iv) (2, 1, 1); 
 
 (v) (1,1,1,1); 
 (vi) PJ'' = 1,QP= 1, Q-'PQ = P'+^\ (2) (11) ; 
 (vii) Pi''=l, Qp = l, R^ = l, R-^QR = QPP, Q-^PQ = P, 
 
 R-^PR = P, (2) (11); 
 (viii) PP' = 1, Qp' = 1, Q~'PQ = P^+P, (11) (11) ; 
 (ix) PP'=1, QP = 1,RP = 1, R-^PR = P'+p, P-^QP = Q, 
 R-'QR = Q, (11) (11), 
 this group (ix) being the direct product of {Q} and {P, R} ; 
 
 (x) Pp'=l, Qp=l, R^ = l, R-'PR = PQ, Q--'PQ = P, 
 R-'QR = Q, (11) (11); 
 
 * On groups of orders p' and p*, the reader may consult, in addition to 
 Young's memoir already referred to, Holder, " Die Gruppen der Ordauugen 
 P'l Pf, Mi'> P*" Math. Ann., xLiir, (1893), in particular, pp. 371—410. 
 
88 TABLES [73 
 
 (xi), (xii), and (xiii) p>3, 
 
 R-'QR = P-pQ, (1)(1)(11), 
 
 where for (xi) a = 0, for (xii) a = 1, for (xiii) a = any non- 
 residue, (mod. p) ; 
 
 (xi), (xii), and (xiii) p = 3, 
 P»=l, Q^ = l, R^=:P^, Q-ipQ = p4^ R-^PR = PQ^, 
 
 R-'QR = Q, (i)(i)(il), 
 
 where for (xi) a = 0, ;S = 1, for (xii) a = 1, /3 = - 1, for (xiii) 
 
 (xiv) PP = 1, QP=1, Rp = l, SP = 1, S-^RS = RP, 
 8-^QS=Q, S-^PS^P, R-^QR = Q, R-^PR = P, 
 Q-^PQ = P, (11) (11), 
 this group (xiv) being the direct product of {Q} and {P, R, >S} ; 
 (xv) p>S, 
 PP = 1, Q»=l, Rp = 1, SP = 1, S-'RS = RQ, S-'QS = QP, 
 S-'PS = P, R-'QR = Q, R-'PR = P, 
 
 Q-^PQ = P, (1)(1)(11); 
 (xv) p = 3, 
 P^=l, Q^=l, R'> = 1, Q-^PQ = P, R-^PR = PQ, 
 R-^QR = P-^Q, (1)(1)(11). 
 
 74. To complete the list, we add, as was promised in § 68, 
 the types of non-Abelian groups of orders 2' and 2^; the 
 possible types of Abelian groups being the same -as for an 
 odd prime. 
 
 Non-Abelian groups of order 2^; two types. 
 
 (i) identical with II (iv), writing 2 for ^ ; 
 
 (ii) P*=l, Q' = l, Q-'PQ = P-\ Q' = P\ (1)(11). 
 Non-Abelian groups of order 2* ; nine types. 
 
 (i), (ii), (iii), (iv) and (v) identical with III (vi), (vii), 
 (viii), (ix) and (x), writing 2 for p ; 
 
75] EXAMPLES 89 
 
 (vi) P' = l, Q>=h R^=l, Qr^PQ = p-\ Q' = F\ 
 R-'QR = Q, R-^PR = P, (11) (11), 
 this group (vi) being the direct product of {R] and {P, Q} ; 
 (vii) P«=l, Q^=l, Qr^PQ = p-\ (1)(1)(11); 
 (viii) P«=l, Q'=l, Q-^PQ = P^, (1)(1)(11); 
 
 (ix) P» = i, Q* = i, Qr'PQ = P-\ Qr = P\ (i)(i)(ii). 
 
 75. In the following examples, G is a non-Abelian group whose 
 
 order is the power of a prime p; and the sub-groups Hi, R.^, , i/!^ 
 
 referred to are the series of self-conjugate sub-groups of § 53. 
 
 Ex. 1. If P and Q are any two operations of G, the totality of 
 operations of the form P~'Q~^PQ generate a self-conjugate sub- 
 group identical with or contained in ff,^. 
 
 G 
 Ex. 2. Shew that, if every sub-group of G is Abelian, -=jr is an 
 
 Abelian group of type (1, 1) ; and any two operations of G, neither 
 of which belongs to H^ and neither of which is a power of the other, 
 generate G. 
 
 Ex. 3. If ^, ^, ^', are of types (1, 1), (1), 
 
 (1), , shew that G can be generated by two operations. 
 
 Ex. 4. If G, of order p™, is not Abelian, and if every sub-group 
 of G is self-conjugate, shew that^j must be 2. (Dedekind.) 
 
 Ex. 5. If G is of order y, and S^ of type (1, 1, with m-1 
 
 units), and if m> ^, G \% the direct product of two groups. If 
 m = t), there is one type for which G is not a direct product, viz. 
 
 QrPQi = P, Q,-'PQ, = P. 
 
 Ex. 6. A group G, of order p™, (m>i), where p is an odd 
 prime, contains an operation P of order ]}'"'"', and no cyclical self- 
 conjugate sub-group of order p™"^. Shew that, if P is one of a set 
 of p^ conjugate operations, the defining relations of the group are of 
 the form 
 
 pp''-'=l, QP=1, Q-' PQ = P'^P""'', R^'^P'^pQP, 
 Ii-^QR=Q, R-'PR = P'*?P'"'"Q; 
 
 and that, if P is one of a set of p conjugate operations, the defining 
 relations are of the form 
 
 Pi'"'-'=1, Qp=l, q-'PQ = P, liP = q'-, 
 
 R-^QB = P?P"'Q, R-^PR = PQ. 
 
 Determine in each case the number of distinct types. 
 
CHAPTER VI. 
 
 ON SYLOWS THEOREM. 
 
 76. It has been proved (§ 22) that the order of any sub- 
 group of a group (? is a factor of the order of G\ and it 
 results at once from the investigation of §§ 4.5 — 47 that, in 
 an Abelian group, there is always at least one sub-group 
 whose order is any given factor of the order of the group. 
 The latter result is not however generally true for groups 
 which are not Abelian ; and for factors of the order of a group 
 which contain more than one distinct prime, no general law is 
 known as to the existence or non-existence of corresponding 
 sub-groups. If however p™, where ^ is a prime, divides the 
 order of the group, it may be shewn that the group will always 
 contain a sub-group of order p""^. The special form of this 
 theorem, that a group whose order is divisible by a prime p 
 contains operations of order p, is due originally to Cauchy*. 
 The more general result was first established by Sylow. He 
 has shewn f that, if ^" is the highest power of a prime p which 
 divides the order of a group, the group contains a single con- 
 jugate set of kp + 1 sub-groups of order p\ 
 
 For the special case 2^-1, the following proof of Cauchy's 
 theorem is perhaps worth giving for its simplicity. Arrange the 
 operations of a group of even order in pairs S and <S^"' of inverse 
 operations. If at any stage no further pairs can be formed, the 
 remaining operations must all, except the identical operation, be 
 
 * Cauchy, Exercises ^analyse, iii, (1844), p. 250. 
 
 + Sylow', Th^or^mes sur les groupes de substitutions. Math. Ann., v (187^), 
 pp. 584 et seq. Compare also Frobenius, Neuer Beweis des Sylow'schen Satzes, 
 Crelle, c. (1886), p. 179. 
 
77] SYLOW'S THEOREM 91 
 
 of order 2. The number of these remaining operations, which 
 include the identical operation, is even. Hence there must be at 
 least one operation of order 2, and the total number of such opera- 
 tions is odd. 
 
 We shall devote the present chapter to the proof of Sylow's 
 theorem ; and a consideration of some of its more immediate 
 consequences. These constitute, as will be seen later on, a most 
 important set of results. 
 
 77. We shall divide the proof of Sylovir's theorem into two 
 parts. First we shew that, if ^" is the highest power of a prime 
 p which divides the order of a group, the group must have a 
 sub-group of order ^"; and secondly that the sub-groups of 
 order p'^ form a single conjugate set and that their number is 
 congruent to unity, mod. p. 
 
 Lemma. If p"^ is the highest power of a prime p by which 
 the order of a groibp Q is divisible, G must contain a sub-group 
 whose order is divisible by p". 
 
 If the group is Abelian, this has been already proved in 
 § 37. Suppose then that G of order N (=p'^m, where m is 
 prime to p) is not Abelian ; and let G' of order N' be the sub- 
 group of G formed of its self-conjugate operations. If S is any 
 operation of G which is not self-conjugate, and if n is the order 
 of the greatest sub-group within which >Si is permutable, S 
 
 N 
 forms one of a set of — conjugate operations; hence, by equa- 
 ting the order of the group to the sum of the numbers of 
 operations contained in the different conjugate sets, we obtain 
 the equation 
 
 N 
 N-==N' + X-, 
 n 
 
 where the sign of summation is extended to all the different 
 conjugate sets of those operations which are not self-conjugate. 
 
 If N' is not divisible by p, or in other words if G contains 
 no self-conjugate operations of order p, this equation requires 
 that at least one of the symbols n should be divisible by p'^ ; 
 therefore in this case the lemma is true. 
 
 If N' is divisible by p"', the group G', being Abelian, must 
 
92 SYLow's [77 
 
 contain a sub-group g' of order p"' ; and this sub-group is self- 
 conjugate in G. Consider now the group — of order p^~'^'m. 
 
 If it has no self-conjugate operations of order p, it must 
 (by the result just obtained) contain a sub-group whose order 
 is divisible byp"""'; and hence Q contains a sub-group whose 
 
 order is divisible by p". If, on the other hand, — has a sub- 
 group of self- conjugate operations of order p"", then G must 
 contain a self-conjugate sub-group of order p"""""", say (/". 
 
 n 
 
 We may now repeat the same reasoning with the group — , of 
 
 order p'^-°-'~'^"m. In this way we must in any case be ultimately 
 led to a sub-group of G whose order is divisible by p^ The 
 lemma is therefore established. 
 
 We have as an immediate inference: — 
 
 Theorem I. //"p" is the highest power of a prime p which 
 divides the order of a group, the group contains a sub-group of 
 order p". 
 
 For the group contains a sub-group whose order is divisible 
 by p" ; this sub-group must itself contain a sub-group whose 
 order is divisible by p", and so on. Hence at last a sub- 
 group must be arrived at whose order is p°- 
 
 Corollary. Since it has been seen in § 53 that a group of 
 order p" contains sub-groups of every order p^ (/3 < a), it follows 
 that, if the order of a group is divisible by p^, the group must 
 contain a sub-group of order p^. In particular, a group whose 
 order is divisible by a prime p has operations of order p. 
 
 78. Theorem II. If p" is the highest power of a prime 
 p which divides the order of a group G, the sub-groups of G of 
 order p" form a single cmijugate set, and their number is con- 
 gruent to unity, mod. p. 
 
 If IT is a sub-group of G of order p^ the only operations of 
 G, which are permutable with H and have powers of p for 
 their orders, are the operations of H itself. For if P is an 
 operation of order p'", not contained in H and permutable with 
 it, and if p' is the order of the greatest group common to [P} 
 
78] THEOREM 93 
 
 and H, the order of {H, P] is p'^+'^-K But can have no such 
 sub-group, since p'^+'^s is not a factor of the order of Q. 
 
 Suppose now that H' is any sub-group conjugate to H ; 
 and let p^ be the order of the group A common to H and H'. 
 When H' is transformed by all the operations of H, the opera- 
 tions of A are the only ones which transform H ' into itself - 
 Hence the operations of U can be divided into p'^~^ sets of p^ 
 each, such that the operations of each set transform H' into a 
 distinct sub-group. In this way, p°-~^ sub-groups are obtained 
 distinct from each other and from H and conjugate to H. If 
 these sub-groups do not exhaust the set of sub-gi'oups conjugate 
 to H, let H" be a new one. From H" another set of p"~^ 
 sub-groups can be formed, distinct from each other and from H 
 and conjugate to H. Moreover no sub-group of this latter set 
 can coincide with one of the previous set. For if 
 
 Pr'H"P, = Pr'H'P„ 
 
 where Pj and P2 are operations of H, then 
 
 H"=PrH'p„ 
 
 where P3 (= PJ^~^) is an operation of H ; and this is contrary to 
 the supposition that H" is different from each group of the 
 previous set. By continuing this process, it may be shewn that 
 the number of sub-groups in the conjugate set containing H is 
 
 l+p''-^+p-'-^'+ , 
 
 where no one of the indices a— y8, a — /S', can be less than 
 
 unity. The number of sub-groups in the conjugate set con- 
 taining H is therefore congruent to unity, mod. p. 
 
 If now contains another sub-group H^ of order p", it must 
 belong to a different conjugate set. The number of sub- 
 groups in the new set may be shewn, as above, to be congruent 
 to unity, mod. p. But on transforming H^ by the operations of 
 H, a set of p^~y conjugate sub-groups is obtained, where py is 
 the order of the sub-group common to H and H^. A further 
 sub-group of the set, if it exists, gives rise to p°-~y' additional 
 conjugate sub-groups, distinct from each other and from the 
 previous p'^~^. Proceeding thus we shew that the number of 
 sub-groups in the conjugate set is a multiple of p ; and as it 
 cannot be at once a multiple of p and congruent to unity, mod. 
 
94 SYLOw's [78 
 
 p, the set does not exist. The sub-groups of order p" therefore 
 form a single conjugate set and their number is congruent to 
 unity, mod. p. 
 
 Corollary I. If p"m is the order of the greatest group I, 
 within which the group H of order p"^ is contained self- 
 conjugately, the order of the group G must be of the form 
 
 p'^m{l +kp). 
 
 Corollary II. The number of groups of order p" contained 
 in G, i.e. the factor 1+kp in the preceding expression for the 
 order of the group, can be expressed in the form 
 
 l+kip + hp^+ ... +Kp'', 
 
 where krp'^ is the number of groups having with a given group 
 H of the set greatest common sub-groups of order p""*". 
 
 This follows immediately from the arrangement of the set 
 of groups given in the proof of the theorem. Thus each of the 
 p<^-? groups, obtained on transforming H' by the operations of 
 H, has in common with H a greatest common sub-group of 
 order p^. It may of course happen that any one or more of the 
 numbers ki, ki, ..., A;^ is zero. If no two sub-groups of the set 
 have a common sub-group whose order is greater than p", then 
 Ai, k^, ..., k„_,._, all vanish ; and the number of sub-groups in the 
 set is congruent to unity, mod. p""^. Conversely, if p^ is the 
 highest power of p that divides kp, some two sub-groups of the 
 set must have a common sub-group whose order is not less than 
 p"'" ; for if there were no such common sub-groups, the number 
 of sub-groups in the set would be congruent to unity, mod. ^*+'. 
 
 Corollary III. Every sub-group of G whose order is p^, 
 (/3<a), must be contained in one or more sub-groups of order p". 
 
 For if the sub-group of order pP is contained in no sub-group 
 of order j3^^S the only operations whose orders are powers of p 
 that transform it into itself are its own. In this case, the 
 preceding method may be used to shew that the number of 
 sub-groups in the conjugate set to which the given sub-group 
 belongs must be congruent to unity, mod. p. But this is 
 impossible, as the number of such sub-groups must, on the 
 assumption made, be a multiple of p'^-^. The sub-group of 
 
79] THEOREM 95 
 
 order ^^ is therefore contained in one of order ^^+S and hence 
 repeating the same reasoning in one of order p"^. 
 
 79. We shall refer to Theorems I and II together as Sylow's 
 theorem. In discussing in this and the following paragraphs 
 some of the results that follow from Sylow's theorem, we shall 
 adhere to the notation that has been used in establishing the 
 theorem itself Thus p" will always denote the highest power 
 of a prime p which divides the order of G ; the sub-groups of G 
 of order ^" will be denoted by H, H^, ..., and the greatest sub- 
 groups of G that contain these selfconjugately by /, /j, .... 
 These latter form a single conjugate set of sub-groups of G, 
 whose orders are p°-m, the order of G itself being p'^mQ. +kp). 
 Moreover the number of groups in this conjugate set is 1 -I- kp. 
 
 Suppose now that S is any operation of G whose order is a 
 power o{p. When the 1 +kp sub-groups 
 
 are transformed by 8, each one that contains S is transformed 
 into itself, while the remainder are interchanged in sets, the 
 number in any set being a power of p. Hence the number of 
 these groups which contain S must be congruent to unity, 
 mod. p. In precisely the same way, it may be shewn that the 
 number of sub-groups of order p°; which contain a given sub- 
 group of order p^, is congruent to unity, mod. p. 
 
 A sub-group (or operation), having a power of ^ for its order 
 and occurring in \+lp sub-groups of order p", will not neces- 
 sarily be one of the same number of conjugate sub-groups (or 
 operations) in each of these 1+lp sub-groups. If then A is a sub- 
 group, of order p^, that occurs in 1+lp sub-groups of order p", 
 we may choose one of these, say H, in which h is one of as small 
 a number as possible of conjugate sub-groups. Let this number 
 be p""*""', so that, in H, h is self-conjugate in a group of order 
 pr+s Then in G, the order of the greatest group i, in which 
 h is self-conjugate, must be p^'^^n, where n is relatively prime to 
 p. The number of sub-groups of i of order p''+* must be 
 congruent to unity, mod. p. No two of these groups of order 
 p^"^' can occur in the same group of order p"^ ; for if they did, 
 they would generate a group of order p''+''+*, {t-^\), and this is 
 
96 DEDUCTIONS FROM [79 
 
 impossible, p''+' being the highest power of p that divides the 
 order of i. Moreover the number of groups of order p" in which 
 any one of these groups of order p''+* enters is congruent to 
 unity, mod. p. Hence the number of groups of order jj", in 
 which h is one of p"-'— « (the least possible number) conjugate 
 sub-groups, is congruent to unity, mod. p. Suppose now that, 
 in H', h is one of p"-*— «' («' <s) conjugate sub-groups; so that 
 in H' it is self-conjugate in a sub-group of order p''+*' and 
 in no greater sub-group. Then the highest power of p that 
 divides the order of the group common to 1' and i is p''+*' ; and 
 therefore, when H' is transformed by all the operations of i, the 
 number of groups of order p" formed is a multiple of p^"^. In 
 each of these groups, h is one of p"-^*' conjugate sub-groups. 
 If this does not exhaust all the groups of order p'^ in which h is 
 one of p---^-^' conjugate sub-groups, let H" be another. Then 
 from this another set of groups, whose number is a multiple of 
 p^~^ , may be formed, which are distinct from each other and 
 from the previous set, such that in -each of them h is one of 
 pa.-r-i conjugate sub-groups. This process may clearly be 
 continued till all such groups are exhausted. Hence the 
 number of groups of order p", in which h is one of p""^""*' (s' < s) 
 conjugate sub-groups, is a multiple of p"'^ . If h enters as one 
 of ^°'-»^*' conjugate sub-groups in H', a sub-group conjugate 
 to h must enter as one of p'^-^—^' conjugate sub-groups in H. 
 Hence if p'^+^, p^^^', jp''+*", ... are the orders of the greatest 
 sub-groups that contain h self-conjugately in the various sub- 
 groups of order p" in which it appears, then H must contain 
 sub-groups of the conjugate set (in G) to which h belongs in 
 conjugate sets of p''+^, p^+^, p'"+«", ... only. The total number 
 X of such sub-groups contained in H is clearly connected with 
 the total number \ + lp of sub-groups of order p'^, in which h 
 appears, by the relation 
 
 x{l+kp) = y{l-^lp), 
 
 y being the number of sub-groups in the conjugate set. For 
 every sub-group of order p"^ will contain x of the set : and the 
 two sides of the equation represent two distinct ways of reckon- 
 ing all the sub-groups of the set A contained in the sub-groups 
 of the set H, when all repetitions are counted. 
 
80] SYLOW'S THEOREM 97 
 
 It is to be noticed that, with the above notation, 
 _ p'^-^^m (1 + kp) 
 ^ ~ n ' ■ 
 
 and therefore x = '^ *- . 
 
 n 
 
 80. Theorem III. Let jp" he the highest power of a prime 
 p which divides the order of a group G, and let H be a sub-group 
 of of order p"^. Let h he a suh-group common to H and some 
 other suh-group of order p"^, such that no sub-group, which contains 
 h and is of greater order, is common to any two sub-groups of 
 order p". Then there must he some operation of G, of order 
 prime to p, which is permutable with h and not with H*. 
 
 Suppose that H and H' are two groups of order p" to 
 which h is common ; and let h^ and A/ be sub-groups of H and 
 H', of greater order than h, in which h is self-conjugate. If h^ 
 and /i]' generate a group whose order is a power of p, it must 
 occur in some group H" of order p°- ; and then H and H" have 
 a common group Ai, which contains h and is of greater order. 
 This is contrary to supposition, and therefore the order of the 
 group generated by h^ and h^ is not a power of p. Hence h is 
 permutable with some operation whose order is prime to p. 
 
 Let p^ be the order of h, and p^+^n be the order of the 
 greatest sub-group i of G that contains h self-conjugately. If i 
 contained a self-conjugate sub-group of order j3''+*, h^ and A/ 
 would be sub-groups of it and they would generate a group 
 whose order is a power of ^. This is not the case, and therefore 
 i must contain 1 -|- h'p sub-groups of order p^'^^, so that we may 
 write m' (1 + h'p) for n ; and then, in i, a sub-group of order p^^" 
 is self-conjugate in a sub-group of order p^+^m'. No sub-group 
 of i of order p'^+* (t > 0) can occur in more than one sub-group 
 of order jj" ; and the 1 -I- k'p sub-groups of i of order p''+* belong 
 therefore to 1 -}- k'p distinct sub-groups of order p"^. Moreover 
 A occurs in no sub-groups of order p" other than these 1 -1- k'p. 
 For if A occurred in another sub-group H^, it would in this 
 sub-group be self-conjugate in a group of order p'^+^ (s' > 0); and 
 
 * Frobenius, " Ueber endliche Gruppen," Berliner Sitzungsberichte (1895), 
 p. 176: and Burnside, "Notes on the theory of groups of finite order," Proc. 
 London Mathematical Society, Vol. xxti (1895), p. 209. 
 
98 DEDUCTIONS FROM [80 
 
 this group would occur in i. This group would then be common 
 to two sub-groups of order p", contrary to supposition. 
 
 An operation of i, which transforms one of its sub-groups 
 of order p''+* into another, must transform the sub-group of order 
 ^" containing the one into that containing the other. Hence i 
 must contain operations which are not permutable with H. 
 The greatest common sub-group of i and / is that sub-group 
 of i of order p'^+^m' which contains the sub-group of order p''+* 
 belonging to H self-conjugately. For every operation, that 
 transforms this sub-group of order p''+* into itself, must trans- 
 form H into itself ; and no operation can transform H into itself 
 which transforms this sub-group of order p''+* into another. 
 
 81. Let P be an operation, or sub-group, which is self- 
 conjugate in H ; and let Q be another operation, or sub-group, 
 of H, which is conjugate to P in G, but not conjugate to P 
 in /. Suppose first that, if possible, Q is self-conjugate in H. 
 There must be an operation 8 which transforms P into Q and 
 H into some other sub-group H', so that 
 
 S-^PS=Q, 
 
 8-'HS=H'. 
 Now in the sub-group which contains Q self-conjugately, 
 the sub-groups of order p" form a single conjugate set, and 
 H must occur among them. Hence this sub-group must contain 
 an operation T such that 
 
 and T-'H'T=H. 
 
 It follows that 
 
 T-^S-'P8T=Q, 
 
 T-'8-'H8T = H, 
 
 or that, contrary to supposition, P and Q are conjugate in I. 
 
 Hence : — 
 
 Theorem IV. Let G and H he defined as in the previous 
 theorem, and let I be the greatest sub-group of G which contains 
 H self-conjugately. Then if P and Q are two self-conjugate 
 operations or sub-groups of H, which are not conjugate in I, 
 they are not conjugate in G. 
 
82] SYLOW'S THEOREM 99 
 
 Corollary. If H is Abelian, no two operations of H which 
 are not conjugate in T can be conjugate in Q. Hence the 
 number of distinct sets of conjugate operations in G, which 
 have powers of p for their orders, is the same as the nvimber of 
 such sets in /. 
 
 82. Suppose next that Q is not self-conjugate in H. 
 Then every operation that transforms Q into P must transform 
 H into a sub-group of order p"^ in which P is not self-conjugate. 
 Of the sub-groups of order p", to which P belongs and in which 
 P is not self-conjugate, choose H' so that, in H', P forms one 
 of as small a number of conjugate operations or sub-groups as 
 possible. Let ^^ be the greatest sub-group of H' that contains 
 P self-conjugately. Among the sub-groups of order jj" that 
 contain P self-conjugately, there must be one or more to which g 
 belongs. Let H be one of these ; and suppose that h and h' 
 are the greatest sub-groups of H and H' respectively that 
 contain g self-conjugately. The orders of both h and h' must 
 (Theorem II, § 55) be greater than the order of g ; and in con- 
 sequence of the assumption made with respect to H', every 
 sub-group, having a power of p for its order and containing li, 
 must contain P self-conjugately. 
 
 Now consider the sub-group {/(, h']. Since it does not 
 contain P self-conjugately, its order cannot be a power of p. 
 Also if ^ is the highest power of p that divides its order, it 
 must contain more than one sub-group of order p^. For any 
 sub-group of order p^, to which h belongs, contains P self- 
 conjugately ; and any sub-group of order p^, to which h' belongs, 
 does not. Suppose now that S is an operation of [h, h'\, having 
 its order prime to p and transforming a sub-group of [h, h'\ of 
 order p^, to which h belongs, into one to which h' belongs. 
 Then S cannot be permutable with P ; for if it were, P would 
 be self-conjugate in each of these sub-groups of order p^. 
 
 When P is an operation, we may reason in the same way 
 with respect to {P}. 
 
 Since g is self-conjugate in both h and h', S must transform 
 g into itself. Now P is self-conjugate in g, and therefore 
 
 7—3 
 
100 SPECIAL TYPES [82 
 
 S~''PS'' is also self-conjugate in g for all values of r. If then 
 S^ is the first power of S which is permutable with P, the 
 
 series of groups P, S'^PS, , S-*+^PS^-'^ are all distinct and 
 
 each is a self-conjugate sub-group of g. Every group in this 
 series is therefore permutable with every other. Hence : — 
 
 Theorem Y. If G and H are defined as in the two preceding 
 theorems, and if P is a self conjugate sub-group or operation of 
 H, then either (i) P must he self-conjugate in every sub-group 
 of G, of order p", in which it enters, or (ii) there must be an 
 operation S, of order q pj-ime to p, such that the set of sub- 
 groups S'^ {P} S^ {r = 0,l .3-1) are all distinct and per- 
 mutable with each other. 
 
 83. In illustration of Sylow's theorem and its consequences, we 
 will now consider certain special types of group ; and we will deal 
 first with a group whose order is the product of two different primes. 
 
 If p^ and p^ (pi < P2) are distinct primes, a group of order 2hlh 
 must, by Sylow's theorem, have a self-conjugate sub-group of order 
 p^. If pnis not congruent to unity, mod. ^1, the group must also 
 have a self-conjugate sub-group of order p^. The two self-conjugate 
 sub-groups of orders p-, and p.2 can ha-\-e no common operation 
 except the identical operation ; and therefore (Theorem IX, § 34) 
 every operation of one must be permutable with every operation of 
 the other. Hence if p.2 is not congruent to unity, mod. p^ , a group 
 of order ^1^0.2 must be Abelian, and therefore also cyclical. 
 
 If p^is congruent to unity, mod. /),, there may be either 1 or p^ 
 conjugate sub-groups of order pj. 
 
 If there is one, it is self-conjugate and the group is cyclical. 
 
 If there are p^ sub-groups of order 2)^ , let Pj denote an operation 
 which generates one of them. Then if P^ is an operation of order p,, 
 
 therefore Pr'AA = -P2% 
 
 so that P2 = P^-PiP,PP' = P,""'. 
 
 Hence a^i = 1 (mod. p^). 
 
 If a were unity, P^ would be permutable with P^, and {P^} would not 
 be one of p^ conjugate sub-groups. Hence a must be a primitive 
 root of the above congruence. 
 
 A group of order PiP^, which has p^ conjugate sub-groups of order 
 Pi , must therefore, when it exists, be defined by the relations 
 
 ppi = l, P/.= l, Pr'P,P, = P, 
 
 2 ! 
 
84] OF GROUP 101 
 
 where a is a primitive root of the congruence ' 
 a^i = 1 (mod. ^j)- 
 
 It follows from § 33 that the order of the group defined by these 
 relations cannot exceed PiP^- But also from the given relations it 
 is clearly impossible to deduce new relations of the form 
 
 -^ 1 -^ 2 — -^ 1 -^2 ) 
 
 where x and x' are less than p^ , and y and y' are less than p^; hence 
 the order cannot be less than iiip^- 
 
 Again, let j8 be a primitive root of the congruence, distinct from 
 a, so that 
 
 ;8'" = a(mod.^2). 
 
 Then if, in the group defined by 
 
 we represent Pi' by P/, the defining relations become 
 P'P \ p i)j _ 1 p '-ip p' .^ p <^ . 
 
 and the group is simply isomorphic with the previous group. 
 
 Hence finally, when p^ is congruent to unity, mod. p-^, there is a 
 single type of group of order p^p^, which contains pa conjugate 
 sub-groups of order p-^. 
 
 84. We will next deal with the problem of determining all 
 distinct types of group of order 24. 
 
 A group of order 24 must contain either 1 or 3 sub-groups of 
 order 8, and either 1 or 4 subgroups of order 3. If it has one 
 sub-group of order 8 and one sub-group of order 3, the group must, 
 since each of these sub-groups is self -conjugate, be their direct 
 product. We have seen (§§ 68, 74) that there are five distinct types 
 of group of order 8 ; there are therefore five distinct types of group 
 of order 24, which are obtained by taking the direct product of any 
 group of order 8 and a group of order 3. 
 
 If there are 3 groups of order 8, some two of them must 
 (Theorem II, Cor. II, § 78) have a common sub-group of order 4 ; 
 and (Theorem III, § 80) this common sub-group must be a self- 
 conjugate sub-group of the group of order 24. Moreover if, in 
 this case, a sub-group of order 8 is Abelian, each operation of the 
 
 ' It should be noticed that, if a is not a root of the congruence, the relations 
 define a group of order pj . Thus from 
 
 ^-^^2^1 = ^2". 
 
 we have p-v, P^ V^, = p/'i . 
 
 Hence P2«'''"'=l, and P/2 = l, 
 
 so that P2 = l, aud the group reduces to {P,}. 
 
102 SPECIAL TYPKS [84 
 
 self-conjugate sub-group of order 4 must (Theorem IV, Cor. § 81) 
 be a self -conjugate operation of the group of order 24. 
 
 With the aid of these general considerations, it now is easy to 
 determine for each type of group of order 8, the possible types of 
 group of order 24, in addition to the five types already obtained. 
 
 (i) Suppose a group of order 8 to be cyclical, and let A be an 
 operation that generates it. If {A\ is self -conjugate and B is an 
 operation of order 3, then 
 
 B-'AB = A'; 
 
 and therefore B-^AW^A"^. 
 
 Hence a^ = 1 (mod. 8), 
 
 and therefore a = 1 (mod. 8) ; 
 
 so that A and B are permutable. This is one of the types already 
 obtained. Hence for a new type, {A\ cannot be self-conjugate, and 
 A^ must be a self -conjugate operation; B is therefore one of two 
 conjugate operations, while {B\ is self-conjugate. Hence the only 
 possible new type in this case is given by 
 
 A-'^BA^B-K 
 
 (ii) Next, let a group of order 8 be an Abelian group defined by 
 
 A''=\, W=\, AB = BA. 
 
 If this is self -conjugate, then, by considerations similar to those 
 of the preceding case, we infer that the group is the direct product 
 of groups of orders 8 and 3. Hence there is not in this case a new 
 type. 
 
 If the group of order 8 is not self-conjugate, the self -conjugate 
 group of order 4 may be either \A\ or {A% B\. In either case, if C 
 is an operation of order 3, it must be one of two conjugate operations 
 while {0} is self -conjugate. Hence there are two new types respec- 
 tively given by 
 
 C'=l, BGB = C-\ A-'CA = C; 
 
 and C'=l, A-^CA = C-', BCB = C. 
 
 (iii) Let a group of order 8 be an Abelian group defined by 
 
 ^"=1, B^=l, 6'^=!, AB = BA, BG=CB, CA=AB. 
 
 If it is self-conjugate, and if the group of order 24 is not the 
 direct product of groups of orders 8 and 3, an operation D of order 3 
 must ti'ansform the 7 operations of order 2 among themselves ; and 
 it must therefore be permutable with one of them. Now the relations 
 
 3-UI) = A, D-^BB = AB, 
 
 are not self-consistent, because they give 
 
 D-WD^ --= B. 
 
84] OF GROUP 103 
 
 Hence, since the group of order 8 is generated hy A, B and any 
 other operation of order 2 except AB, we may assume, without loss 
 of genei'ality, that 
 
 D-^AD = A, D-^BD = C, D-^CD = A^BvC^ 
 
 These relations give 
 
 B = D-^BD^ = D-'^A^'B'JC^D = ^-^ii+^J^^C"*^, 
 
 and therefore y = z= 1. 
 
 Now if D-'CD = ABC, 
 
 and if AB = B', AC=C', 
 
 then £>-'B'D = G', D-'C'D = B'C ; 
 
 so that the two alternatives x = and x=l lead to simply 
 isomoi-phic groups. 
 
 Hence there is in this case a single type. It is the direct 
 product of {A} and {D, B, C\, where 
 
 D-'£D = C, D-'GD = BG. 
 
 If the group of order 8 is not self -conjugate, the self -conjugate 
 group of order 4 may be taken to be [A, B\ ; and D being an opera- 
 tion of order 3, there is a single new type given by 
 
 jy=l, CDC = n-\ ADA = D, BDB=D. 
 (iv) Let a gi'oup of order 8 be a non-Abelian group defined by 
 
 A'=:\, B'=l, A'- = B\ B-'AB = A-^; ^ 
 
 and let G be an operation of order 3. If the group of order 8 is 
 self -conjugate, and the group of order 24 is not a direct product of 
 groups of orders 8 and 3, G must transform the 3 sub-groups of 
 order 4, {A\, [B] and [AB], among themselves. Hence we may take 
 
 C-'AC = B, 
 
 and G-^£G = AB or {ABf. 
 
 If G transforms B into (ABf, then 
 
 C-UG^ = A-\ 
 
 and G cannot be an operation of order 3. Hence in this case there 
 is only one new type, given by 
 
 (?=1, G-'AG = B, G-'BC = AB. 
 
 If the sub-group of order 8 is not self-conjugate, the self-conjugate 
 sub-group of order 4 is cyclical, and eacli of its operations must be 
 permutable wth G. Hence again we get a single new type, given by 
 
 C' = 1, A--'CA = C, B-'GB = G-\ 
 
 (v) Lastly, let a sub-group of order 8 be a non-Abelian group 
 defined by 
 
 ^^ = 1, B- = \, BAB = A-\ 
 
104 SPECIAL TYPES [84 
 
 This contains one cyclical and two non-cyclical sub-groups of 
 order 4. If it is self -conjugate, the group of order 24 must therefore 
 be the direct product of groups of orders 8 and 3 ; and there is no 
 new type. 
 
 If the sub-group of order 8 is not self-conjugate, and the self- 
 conjugate sub-group of order 4 is the cyclical group {A}, then A 
 must be permutable with an operation C of order 3, and there is a 
 single new type given by 
 
 C = 1, A-^CA = C, B-^GB = C-\ 
 If the self-conjugate sub-group of order 4 is not cyclical, it may 
 be taken to be {1, A% B, A^B}. If C is permutable with each 
 operation of this sub-group, there is a single type given by 
 0^=1, A-^OA = C-\ B-^GB=G. 
 If C is not permutable with every operation of the self-conjugate 
 sub-group, it must transform A'', B, A^B among themselves and we 
 may take 
 
 C-'A'G = B, G-'BG = A^B. 
 Now {C, A\ B\ is self-conjugate, and therefore A must transform 
 G into another operation of order 3 contained in this sub-group. 
 Hence 
 
 A-'GA = G'-'A^B''. 
 
 The only values of x, y, z which are consistent with the previous 
 relation 
 
 A'^GA-' = GA-B, 
 are x = 2, y = 2=l. 
 
 ,, The last new type is therefore defined by 
 |; A'=l, 52=1, BAB = A-\ 
 
 I C'=l, G-'AV^B, G-'BG = AW, 
 
 |j A-'GA^G'A^'B. 
 
 I When B is eliminated between these relations, it will be found 
 that the only independent relations remaining are 
 A'=l, G'=l, {AGY=1. 
 
 It is a good exercise to verify that these form a complete set of 
 defining relations for the group. (Compare Ex. 1, § 35.) 
 
 There are therefore, in all, fifteen distinct types of group of order 
 24. The last of these is the only type, which has neither a self- 
 conjugate sub-group of order 8, nor one of order 3. The reader 
 should satisfy himself, as an exercise, that, in the ten cases where 
 the group is not a direct product of groups of orders 8 and 3, the 
 defining relations which we have given are self-consistent. This is 
 of course an essential part of the investigation, and it may, in 
 more complicated cases, involve some little difficulty. We have 
 omitted the verification here, where it is very easy, for the sake of 
 brevity. 
 
85] 
 
 OF GROUP 
 
 105 
 
 It is to be noticed that the last type obtained gives an example, 
 and indeed the simplest possible, of Theorem Y, § 82. Thus in 
 {A, B] of order 8, A^ is a self -conjugate operation and B is not. In 
 the group of order 24, the operations A- and B are conjugate ; and 
 C is an operation, of order prime to 2, such that A^, C~^A'-C, 0~'A^C' 
 generate three mutually permutable sub-groups. 
 
 A discussion similar to that of the present section (but simpler, 
 since in each case the number of types is smaller), will verify the 
 following table* : — 
 
 Order 
 
 6 
 
 10 
 
 12 
 
 14 
 
 15 
 
 18 
 
 20 
 
 22 
 
 26 
 
 28 
 
 30 
 
 Number 
 
 2 
 
 2 
 
 5 
 
 2 
 
 1 
 
 5 
 
 5 
 
 2 
 
 2 
 
 4 
 
 4 
 
 This table, taken with the results of Chapter V, gives the number 
 of distinct types of groups for all orders less than 32, 
 
 85. As a second example, we will discuss the various distinct 
 types of group of order 60. 
 
 A group of order 60 must, by Sylow's theorem, contain either 1 
 or 6 cyclical sub-groups of order 5. 
 
 We will first suppose that a group G of order 60 contains a 
 single cyclical sub-group of order 5, which is necessarily self-con- 
 jugate. There will then be four operations of order 5 in G ; and 
 we may deal with two sub-cases according as these operations are or 
 are not self-conjugate. 
 
 (i) Suppose that each operation of order 5 is self -conjugate. 
 There must then be either 1 or 3 sub-groups of order 4. If there is 
 only one, it must be permutable with an operation of order 3 ; 
 and then G contains a sub-group of order 12. If there are three, 
 it follows, by Theorem II, Cor. II (§ 78), that some pair of them 
 must have a common sub-group of order 2. But (Theorem III, § 80) 
 this sub-group of order 2 must be permutable with some operation 
 of order prime to 2, which is not permutable with a sub-group of 
 order 4. This operation must be of order 3 ; hence in this case also 
 there must be a sub-group of order 12. Thus then the group is, 
 with either alternative, the direct product of a group of order 5 and 
 a group of order 12. Now there are five distinct types of group of 
 order 12; there are therefore five distinct types of group of order 60 
 which contain self -conjugate operations of order 5. 
 
 (ii) Suppose that *S' is an operation of order 5 which is not self- 
 conjugate. If T is an operation which is not permutable with S, 
 then 
 
 T-^ST=S'', 
 * Miller, Gomptcs Rendus, oxxii (1896), p. 370. 
 
106 SPECIAL TYPES 
 
 [85 
 
 where a is not unity. Also, if T" is the lowest power of T which is 
 permutable with iS", then 
 
 and therefore a^ H 1 (mod. 5). 
 
 It follows that X must be either 2 or 4. If a; is 2 for every 
 operation T which is not permutable with S, then S and S^ form a 
 complete set of conjugate operations ; as also do S^ and &^. If x is 
 4 for any operation T, the four operations S, S^, S'\ S* form a 
 single conjugate set. 
 
 First, let S be one of two conjugate operations ; it must then be 
 self -conjugate in a sub-group of order 30, and by Sylow's theorem 
 this sub-group must contain a single sub-group of order 3. It will 
 therefore be given by 
 
 (a) A'=l, 5^ = 1, S'=l, ABA = B, 
 
 or {P) A-'^l, B-'=l, 5= = 1, ABA=B\ 
 
 according as B is or is not a self -con jugate operation; ,S' in either 
 case being permutable with both A and B. 
 
 Jf the sub-groups of order 4 are cyclical, G must contain an 
 operation A^ of order 4 whose square is A ; and A^ must transform S 
 into its inverse and [B] into itself. The latter condition clearly 
 cannot be satisfied if the self-conjugate sub-group of order 30 is of 
 type (/8). Hence we have two types given by 
 
 A^'=l, B' = l, S'=l, B-\^'B = S, 
 
 A,-'SA,:=S* and A-^BA, = B or B'- 
 
 If the sub-groups of order 4 are not cyclical, G must contain an 
 operation A' of order 2, which is permutable with A ; and A' trans- 
 forms a into its inverse and {B} into itself. In this case, if the self- 
 conjugate sub-group is of type (a), there are two types given by 
 
 J'2=l, ^==1, 5^=1, S'=l, ASA=S, B-^SB = S, 
 
 A'AA'^A, A'SA' = S\ ABA = B, A'BA! = BovB\ 
 
 If the self-conjugate sub-group is of type (;8), there is a single 
 type in which the last two of the preceding equations are replaced 
 
 by 
 
 ABA^B\ A'BA'=B. 
 
 Secondly, let the operations of order 5 form a single conjugate 
 set. The sub-groups of order 4 must then be cyclical since G 
 contains an operation T, .such that T* is the lowest power of I' which 
 is permutable with S. Also iS* is permutable in a sub-group of order 
 1 5. This sub-group must be self-conjugate ; and therefore G contains 
 a self-conjugate sub-group of order 3. Let 
 
 B^ = \, S^=\, B-\SB=S 
 
 define the self-conjugate sub-group of order 15 ; and let A^ be an 
 
85] OF Giioup 107 
 
 operation of order 4, none of whose powers is permutable with S. 
 We may then take 
 
 since {B] is self-conjugate, A-^ must transform this sub-group into 
 itself. There are therefore two types given by 
 
 ili^=l, £' = l, S' = l, B-^SB = S, A^-\SA,=S\ 
 
 and A,-'£A, = B or BK 
 
 Hence there are in all twelve distinct groups of order 60, each of 
 which has a self-conjugate sub-group of order 5. 
 
 , (iii) Next, suppose that G contains 6 conjugate sub-groups of 
 order 5. No operation of order 3 can be permutable with an 
 operation of order 5, and therefore by Sylow's theorem there must 
 be 10 conjugate sub-groups of order 3. Hence G contains 24 
 operations of order 5 and 20 operations of order 3. If any one 
 operation of order 5 were permutable with an operation of order 2, 
 all its powers would be permutable with the same operation, and 
 therefore, since the sub-groups of order 5 form a single conjugate 
 set, every operation of order 5 would be permutable with an 
 operation of order 2. The group would then contain at least 24 
 operations of order 10. This is clearly impossible, since the sum of 
 the numbers of operations of orders .3, 5 and 10 would be greater 
 than the order of the group. Hence the sub-group of order 10, 
 which contains self-conjugately a sub-group of order 5, must be of 
 the type 
 
 A^ = l, S'=l, ASA = S*- 
 
 In a similar way, we shew that a sub-group of order 6, which 
 contains self-conjugately a sub-group of order 3, is of the type 
 
 ^==1, 5^ = 1, ABA = B-\ 
 
 Since no operation of order 3 or 5 is permutable with an operation 
 of order 2, it follows (Theorem III, § 80) that no two sub-groups of 
 order 4 can have a common operation other than identity. Hence 
 there must be 5 sub-groups of order 4 ; for if there were 3 or 15, 
 some of them would necessarily have common operations. Each 
 sub-group of order 4 is therefore contained self-conjugately in a 
 sub-group of order 12. Such a sub-group of order 12 can contain no 
 self -conjugate operation of order 2, since G contains no operation of 
 order 6. Hence the sub-groups of order 4 are non-cyclical, and the 
 3 operations of order 2 in any sub-group of order 4 are conjugate 
 operations in the sub-group of order 12 containing it. This sub- 
 group must therefore be of the type 
 
 B'=\, B-'A,B = A,, B-'A,B = A,A,, 
 
 where A^ and A^ are two permutable operations of order 2. 
 
 The 5 sub-groups of order 4 contain therefore 15 distinct opera- 
 tions of order 2 ; and these form a conjugate set. We have already 
 
108 SPECIAL TYPES OF GROUP [85 
 
 seen that the 20 operations of order 3 form a conjugate set and 
 that the 24 operations of order 5 form two conjugate sets of 12 
 each. Hence the 60 operations of the group are distributed in 5 
 conjugate sets, containing respectively 1, 12, 12, 15 and 20 opera- 
 tions. It follows at once (§ 27, p. 35) that the group, when it exists, 
 is simple. 
 
 A sub-group of order 12, the existence of which has been proved 
 must be one of 5 conjugate sub-groups; and, since the group is 
 simple, no operation can transform each of these into itself. Hence 
 if the 5 conjugate sub-groups 
 
 H^, H^, Hg, Bi, //g 
 
 are transformed by any operation of the group into 
 
 if/, H^', Bs, Bi, B^, 
 and if we regard 
 
 Hi, B^, ZTj, B^, B^\ 
 
 Bi, B^, B^, Bl, iZjV 
 
 as a substitution performed on 5 symbols, the group is simply 
 isomorphic with a substitution group of 5 symbols. In other words, 
 the group can be represented as a group of substitutions of 5 symbols. 
 Now there are just 60 even substitutions of 5 symbols ; and it is 
 easy to verify that the group they form satisfy all the conditions 
 above determined. Moreover it will be formally proved in Chapter 
 VIII, and it is indeed almost obvious, that no group of substitutions 
 can be simple if it contains odd substitutions. Hence finally, there 
 is one and only one type of group of order 60 which contains 6 
 sub-groups of order 5. 
 
 Ex. 1. If p, q, r are distinct primes, shew that a group of order 
 jrqr, which contains a self-conjugate operation of order r, must be 
 the direct product of two groups of orders p'q and r respectively. 
 
 Ex. 2. Shew that there is a single type of group of order 84 
 which contains 28 subgroups of order 3 ; and determine its defining 
 relations. 
 
 Ex. 3. If ^j" (a > 1 ) is the highest power of p which divides the 
 order of G, and ii 1 +kp be the number of sub-groups of G of order 
 p", shew that (i) ii 1 + kp <p^, (ii) if a group of order p'- is cyclical 
 and 1 +kp<p''; G is composite. 
 
 (Maillet, Co^nptes Rendus, cxviii, (1894), p. 1188.) 
 
 86. A remarkable and important extension of Sylow's 
 theorem has recently been given by Herr Frobenius'. In 
 the theorem, as stated above, p" is the highest power of a 
 
 ' Frobenius: Berliner Sitzungsherklite {1895), ^. 988. 
 
86] EXTENSION OF SYLOW'S THEOREM 109 
 
 prime p that divides the order of a group. Herr Frobenius 
 shews that, if p'' is any power of a prime p that divides the 
 order of a group, the number of sub-groups of order p'' is 
 congruent to unity, mod. p. These groups do not however, in 
 general, form a single conjugate set. 
 
 For a group whose order is a power of p higher than p'', this 
 result has been already proved in § 61. Suppose now that 
 
 Grk> Gk\ Gk > ) 
 
 are the sub-groups of order p'' contained in a group G, and that 
 p" is the highest power of p dividing the order of 0. It has 
 been seen, in the proof of Sylow's theorem, that G contains at 
 least one sub-group Ga of order p". The above set of sub- 
 groups of order p* may then be divided into two classes, those 
 namely which are contained in (?„, and those which are not. 
 The result of § 61 shews that the number of the sub-groups 
 contained in the first of these classes is congruent to uiiity> 
 mod. p; and if Ga is a self-conjugate sub-group of G, all the 
 sub-groups must be contained in this class. If Ga is not 
 self-conjugate ia G, and if Gk, any one of the sub-groups 
 belonging to the second class, be transformed by all the 
 operations of Ga, a set of p* sub-groups of order p* will result. 
 
 Now it is easy to see that a; is not less than unity, and that 
 every one of these p'' sub-groups belongs to the second class. 
 For suppose first that x is zero, so that Gj is transformed into 
 itself by every operation of Ga. Then [Ga, Gk] is a sub-group 
 of G whose order is a power of ^ ; and since Gk is not contained 
 in Ga, the order of this sub-group is not less than j)"+^ This is 
 impossible, since jo''+' is not a factor of the order of G. Secondly, 
 if P~^GkP were contained in Ga, P being some operation of 
 Ga, then G}c would be contained in PGaP~^ or in Ga, contrary 
 to supposition. If the sub-groups of the second class are 
 not thus exhausted, and if Gk is a new one, we may on trans- 
 forming Gk by the operations of Ga form a fresh set of p'^' 
 sub-groups of the second class which are distinct from each 
 other and from the previous set; and this process may be 
 continued till the second class is exhausted. The number of 
 sub-groups in the second class is therefore a multiple of p. 
 Hence : — 
 
110 FROBENIUS'S 
 
 [86 
 
 Theorem VI. If ]f, where p is prime, divides the order of 
 a grox(,p G, the number of sxib-groups of G of order p* is con- 
 gruent to unity, mod. p. 
 
 87. If p is a prime which divides the order of a group G, 
 it immediately follows from the foregoing theorem that the 
 number of operations of G, which satisfy the relation 
 
 SP = 1, 
 
 is a multiple of p. For the number of sub-groups of G of order 
 p is kip + 1, and no two of these sub-groups can contain a 
 common operation except identity. There are therefore 
 
 {hp + l)(p-l) 
 
 distinct operations of order p in G ; these, together with the 
 identical operation, which also satisfies 
 
 SP=1, 
 
 give in all (k^p — k^+l)p operations. 
 
 This result has been generalized by Herr Frobenius' in the 
 following form : — 
 
 Theorem VII. Ifn is a factor of the order N of a group 
 G, the number of operations of G, including identity, whose 
 orders are factors of n, is a multiple of n. 
 
 It is easy to verify the truth of this theorem directly for 
 small values of N; and we may therefore assume it true for 
 every group whose order is less than that of the given group 0. 
 Again, when n is equal to N, the theorem is obviously true. 
 If then, on the assumption that the theorem is true for all 
 factors of N which are greater than n, we shew that it is true 
 for n, the general truth of the theorem will follow by induction. 
 
 If p is any factor of — , we assume that the number of 
 
 operations of G whose orders divide np is a multiple of np, and 
 therefore also of n ; and we have to shew that the number of 
 operations of G, whose orders divide np and do not divide n, is 
 
 1 " Verallgemeinerung des Sylow'sohen Satzes," Berliner Sitzuvgsberichte 
 (1895), pp. 984, 985. 
 
87] THEOREM 111 
 
 also a multiple of n. Let this set of operations be denoted by 
 A. If j}^~' is the highest power of p that divides n, the order of 
 every operation of the set A must be equal to or be a multiple 
 otpK Let P be one of these operations and m its order. Then, 
 if (f)(m) is the number of integers less than and prime to m, 
 the cyclical sub-group [P] contains <^ (m) operations of order m, 
 and each of these belongs to the set A. If these do not exhaust 
 the set, let P' of order m' be another operation belonging to it. 
 Then no one of the 0(m') operations of order m', contained in 
 the cyclical sub-group {P'}, can be identical with any of the 
 preceding set of (j) (m) operations, since P' itself is not con- 
 tained in that set ; at the same time, the new set of ^ {m') 
 operations all belong to A. This process may be continued till 
 
 the set A is exhausted. Now m, in', are all divisible by ^^, 
 
 and therefore <f) (m), <f> (m'), are all divisible by p''~^ (p — l). 
 
 Hence the number of operations in the set J. is a multiple 
 Of p>'-K 
 
 If now 11 = p''~^s, where by supposition s is relatively prime 
 to p, it remains to shew that the number of operations in the 
 set .4 is a multiple of s. For this purpose, let P be any 
 operation of G of order p'' ; and let those operations of G, which 
 are permutable with P, form a sub-group H of order p'^r. The 
 number of operations of H whose orders divide s is the same as 
 the number whose orders divide t, where t is the greatest common 
 
 TT 
 
 measure of r and s. Now the order of ,-p| is less than N, and 
 
 therefore we may assume that the number of operations of ^p. 
 
 whose orders divide ^ is a multiple of t, say kt. H therefore 
 
 contains kt operations of the form PT, where P and T are 
 
 permutable and the order of T divides s. Now P is one of 
 
 N . 
 a set of -J- conjugate operations in G ; and corresponding to 
 
 each of these, there is a similar set of kt operations. More- 
 over no two of these operations can be identical ; for we have 
 seen (§ 16) that, if m and n are relatively prime, an operation 
 of order mn of a group G, can be expressed in only one way 
 as the product of two permutable operations of G of orders m 
 and n. 
 
112 FROBENIUS'S 
 
 [87 
 
 N 
 The complete set of -^^ kt operations of the form PT belongs 
 
 to ^ ; if J. is not thus exhausted, its remaining operations can 
 be divided into similar sets. Now iV is divisible by both r 
 
 and s, and therefore by their least common multiple — . Hence 
 
 -j^ IS divisible by s, and therefore also the number of opera- 
 tions in the set A is divisible by s. The number of operations 
 in A, being divisible both by ^^-^ and by s, is therefore divisible 
 by n. Hence finally, the number of operations of G, whose 
 orders divide n, is a multiple of n ; and the theorem is proved. 
 
 Corollary I. Let Nn be the number of operations of G 
 whose orders divide n, and suppose that 
 
 Let p be the smallest prime factor of n, and p" the highest 
 power of p that divides v, so that 
 
 where every prime factor of % is greater than p. If the order 
 of is divisible by a higher power of^ than^", then 
 
 Npn = kpn, 
 where k is an integer. Now if the order m of any opera- 
 tion divides pn and does not divide n, then m must be a 
 multiple of ^°+' ; and therefore among the operations, whose 
 orders divide n, there must be operations whose orders are 
 equal to or are multiples of p"^. Hence 
 
 where \ < ^. 
 
 Now iVpa„, — iVj,«-i„j is the number of operations whose orders 
 are factors of n and multiples of p* ; and it has been shewn, in 
 the proof of the theorem, that this number is a multiple of 
 P°~^ (P ~ !)• Hence 
 
 (p — X) ^°~% = yLip""' (^ — 1). 
 Since every prime factor of n^ is greater than p, this equation 
 requires that X should be unity ; therefore 
 
87] THEOREM. 113 
 
 This process may be repeated to shew that, for each value 
 of /3 which is not greater than a, we have 
 
 so that, finally, 
 
 Moreover it is easy to see that the reasoning holds when 
 p'^ is the highest power oip that divides the order of G, provided 
 that then the sub-groups of Q, of order p"^, are cyclical. 
 
 Suppose now that 
 
 n = pj^^P2'^ Pn"", 
 
 where pi, P2, ,Pn are primes in ascending order; and either 
 
 that, for each value of r from 1 to 7i — 1, j)/' is not the highest 
 power of pr which divides the order oi 0: or that, if p,.'^' is the 
 highest power of p^ that divides the order of G, the sub-groups 
 of order |}/' are cyclical. 
 
 Then we may prove as above, first, that G contains opera- 
 tions of each of the orders p^^-^, p^^, , jp^'J^; and secondly 
 
 that, for each value of r from 2 to n, 
 
 N^a. p_a„=p/' PrT". 
 
 The equation 
 
 implies that G has a self-conjugate sub-group of order p^"^; for 
 G has a sub-group of this order, and if G had more than one, it 
 would necessarily contain more than pn"" operations whose 
 orders divide ^„°". 
 
 Again, since G contains a self-conjugate sub-group of order jpn"" 
 and also operations of order p'^^~[, it must contain a sub-group of 
 order p^llPn'- ^^ ^* ^^^ more than one sub-group of this order, 
 it would contain more than p'^"2\p,^ operations whose orders 
 divide p'^"l p "". But since 
 
 this is impossible. Hence G contains a single sub-group of 
 
114 FROBENIUS'S [87 
 
 order p'^ZlPn""' which is necessarily self-conjugate. In the 
 same way we shew that, for each order 
 
 Pr"^ Pn'^(r = n-1, n-2 , 1), 
 
 G contains a single sub-group. 
 
 Finally then, under the conditions stated above, the equa- 
 tion 
 
 involves the property that G has a self-conjugate sub-group of 
 order n and no other sub-group of the same order. 
 
 Corollary II^ If m and n are relatively prime factors of 
 the order of G, and if the numbers of operations of G whose 
 orders divide m and n respectively are equal to m and n ; then 
 every operation whose order is a factor of m is permutable with 
 every operation whose order is a factor of n, and the number of 
 operations of G whose orders divide mn is equal to mn. 
 
 Every operation of G whose order divides mn can be ex- 
 pressed as the product of two permutable operations whose 
 orders divide m and n respectively ; and therefore the number 
 of operations whose orders divide mn cannot be greater than 
 mn. On the other hand, it follows from the theorem that the 
 number of such operations cannot be less than mn ; and therefore 
 every operation whose order divides m must be permutable 
 with every operation whose order divides n. 
 
 Corollary III^. If a group of order mn, where m and n are 
 relatively prime, contains a self-conjugate sub-group of order n, 
 the group contains exactly n operations whose orders divide n. 
 
 If the group G has an operation S whose order divides n, 
 and if S is not contained in the self-conjugate sub-group H of 
 order n, {S, H] would be a sub-group of G, whose order is 
 greater than n and at the same time contains no factor in 
 common with m. This is impossible, and therefore H must 
 contain all the operations of G whose orders divide n. 
 
 • Frobenius, Berliner Sitzungsberichte , (1895), p. 987. 
 
 ^ Frobenius: "Ueber endliche Gruppen," Berliner Sitzungsberichte (1895), 
 p. 170. 
 
88] THEOREM. 115 
 
 Corollary IV'. If has a self-conjugate sub-group H of 
 order mn, where m and n are relatively prime, and if H has a 
 self-conjugate sub-group K of order n, then .fiT is a self-conjugate 
 sub-group of 0. 
 
 For by the preceding Corollary, H contains exactly n opera- 
 tions whose orders divide n, namely the operations forming the 
 sub-group K ; and every operation that transforms H into itself 
 must interchange these n operations among themselves. Hence 
 every operation of Q is permutable with K. 
 
 Ex. 1. Shew that if, the conditions of Corollary II being satis- 
 fied, the order of G is mn, then either G is the direct product of two 
 groups of orders m and n, or G contains an Abelian self -conjugate 
 sub-group. 
 
 Ex. 2. If p and q are distinct primes, there cannot be more than 
 one type of group of order p'^q^ which contains no operation of order 
 pq. 
 
 88. We have seen in | 53 that a group G, whose order is 
 the power of a prime, contains a series of self-conjugate sub- 
 groups 
 
 ■"l) -"2) > -"Jl— 1) -"«) "") 
 
 C H 
 
 such that in ^ every operation of -^^ is self-conjugate. We 
 
 shall conclude the present chapter by shewing that any group 
 which has such a series of self-conjugate sub-groups is the 
 direct product of two or more groups whose orders are powers 
 of primes. 
 
 Theorem VIII. If a group 0, of order p^cf rv, where 
 
 p, q, , r are distinct primes, has a series of self-conjugate 
 
 sub-groups 
 
 ■"1; -"2> -"?!— 1) -"n> ^> 
 
 such that in ^ every operation of -^^ is self-conjugate, then 
 
 is the direct product of groups of order p°-,<f, , r>. 
 
 Suppose, if possible, that p divides the order of H^ and does 
 not divide the order of H^. If P is an operation of order p 
 
 1 Frobenius, loc. cit. p. 170. 
 
 8—2 
 
116 DIRECT PRODUCT OF GROUPS [88 
 
 contained in H^, [P, H^ is a self-conjugate sub-group of 0\ 
 and every operation conjugate to P is contained in the set PH^. 
 But the only operation of this set, whose order is p, is P. Hence 
 P must be a self-conjugate operation, contrary to the suppos- 
 ition that has been made. Hence if the order of E^ is not 
 divisible by p, neither is the order of H^. Suppose, next, that 
 p' is the highest power of p that divides the orders of both H^ 
 
 H G 
 
 and 5^r-i- Then the order of the sub-group — ^ of = — , 
 
 formed of the self-conjugate operations of the latter, is not 
 
 TT 
 
 divisible by p ; and therefore the order of -~^ is not divisible 
 
 -"r-i 
 
 by p. Hence p" is the highest power of p that divides the 
 order of H^+i. This reasoning may be repeated to shew that 
 p" is the highest power oi p that divides the order of each of the 
 
 groups ffr+2, -2r+3> Hence x must be equal to a; and 
 
 therefore the order of H^ must be divisible by each of the 
 primes p, q, , r. 
 
 Suppose now that, for each prime p which divides the order 
 of G, every operation of H^, whose order is a power of p, is 
 permutable with every operation of G whose order is relatively 
 prime to p. Let P be any operation, whose order is a power of 
 p, belonging to -S^+i ^md not to H^ ; and let Q be any operation 
 of G whose order is relatively prime to p. If Q is not per- 
 mutable with P, then 
 
 Q^^PQ^Phr, 
 
 where h^ is some operation of Sr- The order of h^ must be a 
 power of ^. For let h^ = hr'K", where the order of A/ is a power 
 of p and the order of V' is relatively prime to p. Then, from 
 the supposition made with regard to the sub-group Hr, the 
 operation Ph^ is the product of the permutable operations PV 
 and hr". But, since the order of PK is a power of p, this 
 is impossible unless h" is identity. If the order of h^ is j)^, 
 then 
 
 (^ PQ?^ = Ph/ = P ; 
 
 and this equation implies that Q is permutable with P, since 
 p^ and the order of Q are relatively prime. Hence if the 
 
88] WHOSE ORDERS ARE POWERS OF PRIMES. 117 
 
 supposition that has been made holds for H^, it also holds for 
 Er+i- But it certainly holds for H^, and therefore it is true for 
 0. Hence every operation of G' whose order is a power of p is 
 permutable with every operation of G whose order is relatively 
 prime to p. The group therefore contains self-conjugate sub- 
 groups of each of the orders^", q^, , rf ; and it follows, from 
 
 the definition of § 31, that is the direct product of these 
 groups. 
 
 We add here two examples in further illustration of the applica- 
 tions of Sylow's theorem. 
 
 Ex. 1. li p is & prime, greater than 3, shew that the number of 
 distinct types of group of order &p is 6 or 4, according as p is con- 
 gruent to 1 or 5, mod. 6. 
 
 Ex. 2. If ^ is a prime, greater than 5, shew that the number of 
 distinct types of group of order 12p is 18, 12, 15 or 10, according 
 as^ is congruent to 1, 5, 7 or 11, mod. 12. 
 
CHAPTER VII. 
 
 ON THE COMPOSITION-SERIES OF A GROUP. 
 
 89. Let (ti be a maximum self-conjugate sub-group (§ 27) 
 of a given group Q, 0^ a maximum self-conjugate sub-group of 
 G^i, and so on. Since G* is a group of finite order, we must, after 
 a finite number of sub-groups, arrive in this way at a sub-group 
 Gn~i, whose only self-conjugate sub-group is that formed of the 
 identical operation alone, so that (t„_i is a simple group. 
 
 Definitions. The series of groups 
 
 G, Gi, G~i, , &„_!, 1, 
 
 obtained in the manner just described is called a composition- 
 series of G. 
 
 The set of groups 
 
 /~r? /nf> i ri j ^n— 1 j 
 
 ITi (Ta Cr,i_i 
 
 is called a set o^ factor-groups of G, and the orders of these 
 groups are said to form a set of composition-factors of G. 
 
 Each of the set of factor-groups is necessarily (§ 30) a 
 simple group. 
 
 The set of groups forming a composition-series of G is not, 
 in general, unique. Thus G may have more than one maximum 
 self-conjugate sub-group, in which case the second term in the 
 series may be taken different from Gi. Moreover the groups 
 succeeding Gi are not all necessarily self-conjugate in G; and 
 when some of them are not so, we obtain a new composition- 
 
90] COMPOSITION SERIES. 119 
 
 series on transforming the whole set by a suitably chosen 
 operation of 0. That the new set thus obtained is again a 
 composition-series is obvious ; for if Or+\ is a maximum self- 
 conjugate sub-group of Gr, so also is S~^Gr+iS o{ S~^Qr8. We 
 proceed to prove that, if a group has two different composition - 
 series, the number of terms in them is the same and the 
 factor-groups derived from them are identical except as regards 
 the sequence in which they occur. 
 
 This result, which is of great importance in the subsequent 
 theory, is due to Herr Holder'; and the proof we here give does 
 not differ materially from his. 
 
 The less general result, that, however the composition-series 
 may be chosen, the composition-factors are always the same 
 except as regards their sequence, had been proved by M. Jordan^ 
 some years before the date of Herr Holder's memoir. 
 
 90. Theorem I. If H is any self-conjugate sub-group of a 
 group G; and if K, K' are two self -conjugate sub-groups of G con- 
 tained in H, such that there is no self -conjugate sub-group of G 
 contained in H and containing either K or K' except H, K and 
 K' themselves; and if L is the greatest common sub-group of K 
 and K', so that L is necessarily self-conjugate in G; then the 
 
 groups -j^ and -^ are simply isomorphic, as also are the groups 
 
 H . K 
 j^i and -J . 
 
 Since K and K' axe self-conjugate sub-groups of G contained 
 in H, [K, K'} must also be a self-conjugate sub-group of G 
 contained in H ; and since, by supposition, there is in H no self- 
 conjugate sub-group of G other than H itself, which contains 
 either K or K', [K, K'] must coincide with H. Hence (§ 38) 
 the product of the orders of K and K' is equal to the product 
 of the orders of H and L. 
 
 If the order of -^ is m, the operations of K may be divided 
 
 into the m sets 
 
 L, SiL, 82L, , Sjn^iL, 
 
 ' " Zuriickfiihrung einer beliebigen algebraischen Gleiohung auf eine Kette 
 von Gleiohungen," Math. Ann. xxxiv, (1889), p. 33. 
 * "Traits des substitutions," (1870), p. 42. 
 
120 COMPOSITION [90 
 
 such that any operation of one set multiplied by any operation 
 of a second gives some operation of a definite third set, and the 
 
 group y is defined by the laws according to which the sets 
 
 combine. 
 
 Consider now the m sets of operations 
 
 K , biK', S2K', , Sm-iK'. 
 
 No two operations of any one set can be identical. If 
 operations from two different sets are the same, say 
 
 where k-^ and k^ are operations of K', then 
 
 ^q ^p ^^ '^2 '^l ) 
 
 some operation of K'. But 8q~^Sp is an operation of K ; hence, 
 as it belongs both to K and K', it must belong to L, so that 
 
 Sp = Sql, 
 
 where I is some operation of L. This however contradicts the 
 supposition that the operations SpL and SqL are all distinct. It 
 follows that the operations of the above m sets are all distinct. 
 
 Now they all belong to the group H ; and their number, 
 
 being the order of K' multiplied by the order of 7^, is equal to 
 
 the order of H. Hence in respect of the self-conjugate sub- 
 group K', which H contains, the operations of the group H can 
 be divided into the sets 
 
 K', SiK', 82K', , Sm-iK', 
 
 JT 
 
 and the group ^ is defined by the laws according to which 
 these sets combine. But if 
 
 then necessarily 
 
 8pK'.8qK' = 8rK'. 
 
 H K 
 
 Hence the groups -^, and y are simply isomorphic. In pre- 
 
 77 K' 
 
 cisely the same way it is shewn that -^ and y are simply 
 
 isomorphic. 
 
91] SERIES. 121 
 
 Corollary. If H coincides with G, K and K' are maximum 
 
 self-conj ugate sub-groups of G. Hence if K and K' are maximum 
 
 self-conjugate sub-groups of G, and if L is the greatest group 
 
 C K' 
 
 common to K and K', then ^ and -^ are simply isomorphic; as 
 
 also are -^ and y . 
 
 C G K K' 
 
 Now -j^ and -f^, are simple groups ; and therefore, -^ and —jr- 
 
 K. K. h Li 
 
 being simple groups, L must be a maximum self-conjugate 
 sub-group of both K and K'. 
 
 91. We may now at once proceed to prove by a process of 
 induction the properties of the composition-series of a group 
 stated at the end of § 89. Let us suppose that, for groups 
 whose orders do not exceed a given number n, it is already 
 known that any two composition-series contain the same 
 number of groups and that the factor-groups defined by them 
 are the same except as regards their sequence. If G, a group 
 whose order does not exceed 2n, has more than one composition- 
 series, let two such series be 
 
 G, Gi, ©2, , 1 ; 
 
 and G,G,',G,', 1. 
 
 If H is the greatest common sub-group of Gi and Gi, and if 
 
 H,I,J, 1 
 
 is a composition-series of H, then, by the Corollary in the 
 preceding paragraph, 
 
 G, Gi, H, I, J, , 1, 
 
 and G,G,',H,I,J, , 1 
 
 are two composition-series of G which contain the same number 
 
 of terms and give the same factor-groups. For it has there 
 
 p pi 
 
 been shewn that jy- and -^ are simply isomorphic ; as also are 
 
 p-, and -^ . Now the order of Gi, being a factor of the order of 
 
 G, cannot exceed n. Hence the two composition-series 
 
 Cti, Gti, , 1, 
 
 aiid G„H,I, ,1, 
 
122 COMPOSITION SERIES. [91 
 
 by supposition contain the same number of groups and give the 
 same factor-groups ; and the same is true of the two composition- 
 series 
 
 G\, Gi> 1, 
 
 and G-:,H,I, 1. 
 
 Hence finally, the two original series are seen, by comparing 
 them with the two new series that have been formed, to have 
 the same number of groups and to lead to the same factor- 
 groups. The property therefore, if true for groups whose order 
 does not exceed n, is true also for groups whose order does not 
 exceed 2w. Now the simplest group, which has more than one 
 composition-series, is that defined by 
 
 A^ = \, & = \, AB = BA. 
 
 For this group there are three distinct composition-series, viz. 
 
 [A,B},{A}, 1: 
 
 [A,B},[B], 1 
 
 and {A,B], [AB], 1: 
 
 and for these the theorem is obviously true. It is therefore 
 true generally. Hence : — 
 
 Theorem II. Any two composition-series of a group consist 
 of the same number of sub-groups, and lead to two sets of factor- 
 groups which, except as regards the sequence in which they occur, 
 are identical with each other. 
 
 The definite set of simple groups, which we thus arrive at 
 from whatever composition-series we may start, are essential 
 constituents of the group : the group is said to be compounded 
 from them. The reader must not, however, conclude either that 
 the group is defined by its set of factor-groups, or that it necess- 
 arily contains a sub-group simply isomorphic with any given 
 one of them. 
 
 92. It has been already pointed out that the groups in a 
 composition-series of G are not necessarily, all of them, self- 
 conjugate groups of G. 
 
 Suppose now that a series of groups 
 
 G, Hi, H2, , Hm-i, 1 
 
92] CHIEF SERIES. 123 
 
 are chosen so that each one is a self-conjugate sub-group of G, 
 while there is no self-conjugate sub-group of G contained in any 
 one group of the series and containing the next group. 
 
 Definition. The series of groups, obtained in the manner 
 just described, is called a chief composition-series, or a chief- 
 series of G- 
 
 It should be noticed that such a series is not necessarily 
 obtained by dropping out from a composition-series those of its 
 groups which are not self-conjugate in the original group. 
 Thus it follows immediately, from the results of § 54, that the 
 composition-series of a group whose order is the power of a 
 prime can be chosen, either (i) so that every group of the series 
 is a self-conjugate sub-group, or (ii) so as to contain any given 
 sub-group, self-conjugate or not. 
 
 A chief composition-series of a group is not necessarily 
 unique; and when a group has more than one, the following 
 theorem, exactly analogous to Theorem II, holds : — 
 
 Theorem III. Any two chief -composition-series of a group 
 consist of the same number of term,s and lead to two sets of 
 factor-groups, which, except as regards the sequence in which they 
 occur, are identical with each other. 
 
 The formal proof of this theorem would be a mere repetition 
 of the proof of § 91, Theorem I itself being used to start from 
 instead of its Corollary ; it is therefore omitted. 
 
 Although it is not always possible to pass from a composition- 
 series to a chief-series, the process of forming a composition- 
 series on the basis of a given chief-series can always be carried 
 out. Thus if, in a chief-series, -ffr+i is not a maximum self- 
 conjugate sub-group of Hr, the latter group must have a 
 maximum self-conjugate sub-group Gr,i which contains Hr+i- 
 If Hr+i is not a maximum self-conjugate sub-group of Gr,i, then 
 such a group, Gr^^, may be found still containing Hr+i ; and this 
 process may be continued till we arrive at a group G^.s-i, of 
 which Hr+i is a maximum self-conjugate sub-group. A similar 
 process may be carried out for each pair of consecutive terms 
 m the chief-series ; the resulting series so obtained is a com- 
 position-series of the original group. 
 
124 MINIMUM [93 
 
 93. The factor-groups „- ^ arising from a chief-series are 
 
 not necessarily simple groups. If between H^ and iT^+i no 
 groups of a corresponding composition-series occur, the group 
 
 TT 
 
 jj- ^ is simple ; but when there are such intermediate groups, 
 
 ZJ" 
 
 Tj- ^ cannot be simple. We proceed to discuss the nature of 
 this group in the latter case. 
 
 Let G be multiply isomorphic with G', so that the self- 
 conjugate sub-group Hr+i of G corresponds to the identical 
 operation of G'. Also let 
 
 Hi, H2, , Hp, H'p^i, Hr', 1 
 
 be the sub-groups of G' which correspond to the sub-groups 
 
 of G. Since Hp contains Hp+i, Hp' must contain H'pj^^; and 
 since jET^+i is self-conjugate in G, S'p+i is self-conjugate in G'. 
 Also if G' had a self-conjugate sub-group contained in Hp' and 
 containing -ff'j,+i, G would have a self-conjugate sub-group 
 contained in Hp and containing Hp^^. This is not the case, 
 and therefore 
 
 G', Hi, Hi, , Hr, 1 
 
 TT 
 
 is a chief-series of G'. Hence ^ *" is simply isomorphic with 
 Hr, the last group but one in the chief-series of G' . 
 
 Definition. If F is a self-conjugate sub-group of G, and 
 if G has no self-conjugate sub-group, contained in V, whose 
 order is less than that of F, then F is called a minimum self- 
 conjugate sub-group of G. 
 
 Making use of the phrase thus defined, the discussion of the 
 
 factor-groups '" of a chief-series is the same as that of the 
 
 -"r+i 
 
 minimum self-conjugate sub-groups of a given group. 
 
 94. To simplify the notation as much as possible, let / be 
 a minimum self-conjugate sub-group of G ; and, if I is not a 
 simple group, suppose that in a composition-series of G the 
 
94] SELF-CONJUGATE SUB-GROUP. 125 
 
 term following I is g^. Since g-^ cannot be self-conjugate in G, 
 it must be one of a set of conjugate sub-groups 
 
 g-^,92, ,gn («)• 
 
 Now, if gr = S~^giS, then as g'l is a maximum self- conjugate 
 sub-group of /, 8~'^giS or gr is a maximum self-conjugate 
 sub-group of 8~^IS or / ; and hence every one of the above set 
 of conjugate sub-groups is a maximum self-conjugate sub-group 
 of /. If grs represents the greatest sub-group common to g,. and 
 9s, then, by Theorem I of the present chapter (§ 90), 
 
 ,I,gi,9ir, 
 
 ) -' J gr> gin 
 
 are composition-series of G. Hence - is simply isomorphic 
 
 9ir 
 
 with — . Now — and „ — ^r-, , or — , are simply isomorphic ; 
 g, gr SgrS-"- g, 
 
 therefore — and — are simply isomorphic. Similarly we may 
 
 fflr 9l 
 
 shew that, whatever r and s may be, — and — are simply 
 
 9rs gi 
 
 isomorphic. 
 
 If gir and g^g are the same group, whatever r and s 
 may be, this group is common to the whole set of conjugate 
 groups (a). If these groups have any common sub-group, 
 except identity, it is (Theorem V, § 27) a self-conjugate sub- 
 group of G, and this self-conjugate sub-group would be con- 
 tained in /, contrary to supposition. Hence if gir and g^g are 
 the same group, for all values of r and s, this group consists of 
 the identical operation alone, and a composition-series is 
 
 .-f, 5'i. 1- 
 
 If 5'i,. and g^g are distinct and if ^Tj^^ is their greatest common 
 sub-group, then 
 
 J -') 9l> 9ir! 9irsy 
 
 ! -^ > 9i> 9is > 9irs > 
 
 are two composition-series, and — is simply isomorphic with - 
 
 9irs 9i 
 
 for all values of r and s. 
 
126 MINIMUM [94 
 
 This reasoning may be repeated. If ^i„ and g^rt are the 
 same group whatever s and t may be, they must each be the 
 identical operation. If not, g^rst is another term of the composi- 
 tion-series, and ^— is simply isomorphic with - . 
 9irst gi 
 
 Hence however the composition-series from / onwards be 
 constructed, the corresponding factor-groups are all simply 
 isomorphic with each other. Moreover every group after I in 
 the composition-series is self-conjugate in /. For g^ and g^ 
 being self-conjugate sub-groups of I, so also is their greatest 
 common sub-group g^r', and g^,., gis being self-conjugate sub- 
 groups of /, g^rs is also self-conjugate; and so on. Let the 
 composition-series thus arrived at be now written 
 
 , -^, Ti' 72 Js-i, 1 ; 
 
 where, as has been proved, — , — , , <ys-i are simply isomorphic. 
 
 The final group 7j_i must be one of a conjugate set of, say, 
 V groups in 0, no one of which has any operation except 
 identity in common with any other. Since each of these v 
 groups is self-conjugate in /, and since no two of them have a 
 common operation except identity, it follows by Theorem IX, 
 § 34, that every operation of any one of them is permutable 
 with every operation of the remaining v—1. The group gen- 
 erated by the p groups conjugate to ys-i> being self-conjugate in 
 G and contained in /, must coincide with /. Now, if 7\_i and 
 7V1 are any two of this set of v groups, {7^-1, 7V1} is their 
 direct product, and it is a self-conjugate sub-group of I. If 
 s > 2, {7^-1, 7\-i} does not coincide with I; and therefore there 
 must be another sub-group rfs-i> of the set to which ^s-i 
 belongs, which is not contained in {7^-1, 7%-i}- Since both 
 the latter group and 7^s_i are self-conjugate in /, while 7^g_i 
 is a simple group, no operation of 7's_i except identity can be 
 contained in {7^8-1, y's-i}- Hence y^s-i> 'y's-i and 7V1 are inde- 
 pendent, i.e. no operation of one of these groups can be ex- 
 pressed in terms of operations of the other two. The group 
 {7's-i. 7V1, 7's-i} is the direct product of 7's_i, 7V1 and 7*j_i, 
 and it is self-conjugate in /. If s > 3, the same reasoning 
 
95] SELF-CONJUGATE SUB-GROUP. 127 
 
 may be repeated. Finally, from the set of v groups conjugate 
 to 7j_i , it must be possible to choose s independent groups 
 
 7'«-i. 7V1. , 7'.-i. 
 
 such that no operation of any one of them can be expressed in 
 terms of the operations of the remaining s — 1 ; and I will then 
 be the direct product of these s groups. Hence : — 
 
 Theorem IV. If between two consecutive terms Hy and Hr+i 
 in the chief-composition-series of a group there occur the groups 
 
 Gr,i, Gr,2, , Gr,e-i of a composition-series ; then (i) the factor 
 
 groups 
 
 ■tir "r, 1 "r, 5—1 
 
 7> > p ' ' "jCT 
 
 ^r, 1 ^r, 2 -" r+1 
 
 TT 
 
 are all simply isomorphic, and (ii) „ ^ *^ the direct product of 
 
 Hr+l 
 
 s groups of the type j^ . 
 
 TT 
 
 Corollary. If the order of „ ' is a power, p', of a prime, 
 
 -"r+l 
 
 TT 
 
 jY^ must be an Abelian group of type (1, 1 to s units). 
 
 95. A chief-series of a group G can always be constructed 
 
 which shall contain among its terms any given self-conjugate 
 
 sub-group of G. For if F is a self-conjugate sub-group of G, 
 
 C 
 and if ^ is simple, we may take F for the group which follows 
 
 C 
 G in the chief-series. If on the other hand -^ is not simple, it 
 
 must contain a minimum self-conjugate sub-group. Then Fi, 
 the corresponding self-conjugate sub-group of G, contains F ; 
 and if there were a self-conjugate sub-group of contained in 
 
 Fi and containing F, the self-conjugate sub-group of = , which 
 
 corresponds to Fj, would not be a minimum self-conjugate sub- 
 group. We may now repeat the same process with Fj, and so 
 on; the sub-groups thus introduced will, with G and F, 
 clearly form the part of a chief-series extending from G to F. 
 The series may be continued from F, till we arrive at the 
 identical operation, in the usual way. 
 
1 28 EXAMPLES. [96 
 
 96. It will perhaps assist the reader if we illustrate the fore- 
 going theory by one or two simple examples. We take first a group 
 of order 12, defined by the relations 
 
 A^=l, £^=1, AB = BA, 
 
 B^^l, R-^AR = B, R-^BR = AB*. 
 
 From the last two equations, it follows that 
 
 R-'^ABR^A, 
 
 and therefore R transforms the sub-group [A, B] of order 4 into 
 itself ; so that this sub-group is self-conjugate, and the order of the 
 group is 12 as stated. The self -conjugate sub-group [A, B\ thus 
 determined is clearly a maximum self-conjugate sub-group. Also it is 
 the only one. For if there were another its order would be 6, and 
 it would contain all the operations of order 3 in the group. Now 
 since [R] is only permutable with its own operations, the group 
 contains 4 sub-groups of order 3, and therefore there can be no 
 self-conjugate sub-group of order 6. The three cyclical sub-groups 
 {^}> {^] ^iid {AB] of order 2 are transformed into each other by R, 
 and therefore no one of them is self-conjugate. 
 
 Hence the only chief-series is 
 
 {R,A,B}, {A,B}, 1, 
 
 and there are three composition-series, viz. 
 
 {R, A, B], 
 
 {A, B], 
 
 {A}, 1 : 
 
 {R, A, B], 
 
 {A, B], 
 
 [B], 1: 
 
 {R, A, B], 
 
 {A, B], 
 
 [AB], 1. 
 
 and 
 
 The orders of the factor-groups in the chief-series are 3 and 2^, 
 and the group of order 2^ is, as it should be, an Abelian group whose 
 operations are all of order 2. The composition-factors are 3, 2, 2 in 
 the order written. 
 
 97. As a rather less simple instance, we will now take a group 
 generated by four permutable independent operations A, B, P, Q; of 
 orders 2, 2, 3, 3 respectively and an operation R of order 3, for 
 which 
 
 R-^AR = B, R-^BR = AB, R-'PR^F, R-'^QR^QPi. 
 
 The sub-group {A, B, P, Q}, of order 36, is clearly a maximum 
 self-conjugate sub-group, and therefore the order of the group is 108. 
 Since A, B and AB are conjugate operations, every self -conjugate 
 sub-group that contains A must contain B ; and since Q and QP are 
 
 * The reader will notice that B can be eliminated from these relations, and 
 that the group can be defined by ^'=1, 11^=1, {AR)^=:1. The structure of the 
 group however is given, at a glance, by the equations in the text. 
 
 t Here again the group can clearly be defined in terms of ^, Q and R. 
 
98] OF COMPOSITION SERIES 129 
 
 conjugate, every self-conjugate sub-group that contains Q must 
 contain P. Hence the only other possible maximum self -conjugate 
 sub-groups are those of the form {A, B, P, EQ"'} ; and since 
 
 these groups actually are self-conjugate. The same reasoning shews 
 that the only maximum self-conjugate sub-group of {A, B, P, Q} or 
 of [A, B, P, EQ"}, which is self -conjugate in the original group, is 
 \A, B, P} ; and the only maximum self-conjugate sub-groups of the 
 latter, which are self-conjugate in the original group, are {A, B} and 
 {P}. Hence all the chief-series of the group are given by 
 {A, B, P, Q], {A, B], 
 
 {E,A,B,P,Q}, or {A, B, P}, or 1. 
 
 {A, B, P, EQ% {P}, 
 
 Since {A, B, P, Q} is an Abelian group, all of its sub-groups are 
 self-conjugate. Hence if G^, G^ and G^ are any maximum sub-groups 
 of {A, B, P, Q}, <?! and G^ respectively, then 
 
 {B, A, B, P, Q], {A, B, P, Q}, G„ G„ G„ 1 
 
 is a composition-series. 
 
 Again, since A and B are conjugate in {A, B, P, EQ°'}, the only 
 maximum self -conjugate sub-groups of this group are those of the 
 form {A, B, P^ [EQ'^y]. If y is zero, this sub-group is Abelian ; and 
 we may take for the next term in the composition series any 
 maximum sub-group g^ of this Abelian sub-group, and for the last 
 term but one any sub-group g^ of g^. But if y is not zero, 
 {A, B, P''{EQy) can only be followed by {A, B}. Hence the 
 remaining composition-series are of the forms : — 
 
 {R,A,B,P,Q}, {A, B, P, RQ'^], {A, B, P], g„ g„ 1, 
 and 
 
 {E,A,B,P,Q}, {A,B,P,RQ\ {A, B, P^iEQy}, 
 {A,B}, {A} or {B} or {AB}, 1. 
 
 It should be noticed that if, in the last of these series, we drop out 
 the terms which are not self-conjugate in the original group, here 
 the third and fifth terms, we do not arrive at a chief-series. This 
 illustrates a remark made in § 92. 
 
 98. Theorem V. If H is a sub-group of G, each com- 
 positionfactor of If must be equal to or be a factor of some 
 composition-factor of 0. 
 
 If G is simple, its only composition-factor is equal to its 
 order : the theorem in this case is obvious. 
 
130 SOLUBLE [98 
 
 If G is not simple, let Gr+i be the first term in a com- 
 position-series of G which does not contain H; and let Gr be 
 the term preceding G^r+i- If H^ is the greatest common sub- 
 group of G?,.+i and H, then H^ is a self-conjugate sub-group 
 of H. For every operation of H transforms both H and G^^j 
 into themselves ; and therefore every operation of H transforms 
 H^, the greatest common sub-group of H and (?r+i, into itself 
 Now the order of {H, Gr+i] is equal to the product of the 
 orders of iT and Gr+^. divided by the order of ^i; and [H, Gr+^ 
 
 rr 
 
 is contained in Gy Hence the order of w is equal to or is 
 
 a factor of the order oi^-- . If then a composition-series of H 
 
 be taken, containing the term H^, the orders of the factor- 
 groups, formed by those terms of the series terminating with 
 
 H^, are equal to or are factors of the order of -pr~- The same 
 
 reasoning may now be used for Hx that has been applied to H; 
 and the theorem is therefore true. 
 
 Corollary. If all the composition-factors of G are primes, 
 so also are the composition-factors of every sub-group of G. 
 
 99. Definition. A group, all whose composition-factors 
 are primes, is called a soluble group. 
 
 The Corollary of the last theorem may be stated in the 
 form : — if a group is soluble, so also are all its sub-groups. 
 
 A soluble group of order p'^q^ rv, where p, q , r 
 
 are distinct primes, has a + yS-1- +<y composition-factors ; 
 
 these are capable of ^^ , „, — r^ distinct arrangements. 
 
 ^ a!/3! 7! 
 
 For a specified group the composition-factors may, as we 
 have already seen, occur in two or more distinct arrangements ; 
 but it is immediately obvious that two groups of the same 
 order cannot be of the same type, i.e. simply isomorphic, unless 
 the distinct arrangements, of which the composition-factors are 
 capable, are the same for both. A first step therefore towards 
 the enumeration of all distinct types of soluble groups of a 
 given order, will be to classify them according to the distinct 
 
100] GROUPS 131 
 
 arrangements of which the composition-factors are capable ; for 
 no two groups belonging to different classes can be of the same 
 type. 
 
 The case, in which the composition-factors are capable of all 
 possible arrangements, is one which will always occur. Taking 
 in this case /3 q's followed by a ^'s for the last a + composition- 
 factors, the group contains a sub-group G' of order p'^q^. In 
 the composition-series of this group, with the composition- 
 factors taken as proposed, there is a sub-group H of order p" 
 contained self-conjugately in a sub-group H^ of order p^q. This 
 sub-group Hi is contained self-conjugately in a group H2 of 
 order p^q^ Hence (Theorem VII, Cor. IV, § 87) H is 
 contained self-conjugately in H^. Again, H2 is contained self- 
 conjugately in a group H^ of order p'^q^, and therefore again H 
 is self-conjugate in ZTj. Proceeding thus, we shew that H is 
 self-conjugate in G'^ It follows that n, the number of 
 conjugate sub-groups of order p' contained in the group, is 
 not a multiple of q. Now q may be any one of the distinct 
 primes other than p that divide the order of the group. Hence 
 finally the group contains a self-conjugate sub-group of order 
 p". In the same way we shew that it contains self-conjugate 
 
 sub-groups of orders q^, , r"". The group must therefore 
 
 be the direct product of groups whose orders are p", q^, , r''. 
 
 Hence : — 
 
 Theorem VI. A soluble group, the composition-factors 
 of which may be taken in any order, is the direct product of 
 groups whose orders are powers of primes. 
 
 100. Theorem VII. If G is a soluble group of order p'^m, 
 where p is prime and does not divide m, and if every operation 
 of G whose order is a power of p is permutable with every 
 operation whose order is relatively prime to p ; then G is the 
 direct product of two groups of orders p" and m. 
 
 Let if be a sub-group of G of order j)". This sub-group, 
 from the conditions of the theorem, is necessarily self-conjugate. 
 Let a chief-series of G be formed which contains H, say 
 
 G, H',H, , 1. 
 
132 SOLUBLE GROUPS 
 
 [100 
 
 The order of H' must be of the form p-'q^ ; and H' must 
 contain a single sub-group h of order /. This sub-group h is 
 self-conjugate in G, since H' is self-conjugate in G; and h is the 
 only sub-group of order q^ contained in H'. 
 
 Consider now the group -y-. Every operation of this group 
 
 whose order is a power of p is permutable with every operation 
 
 whose order is relatively prime to p. Hence we may repeat 
 
 C 
 the same reasoning to shew that -j- contains a self-conjugate 
 
 sub-group of order rv, where r is a prime distinct from p, but 
 possibly the same as q. It follows that G has a self-conjugate 
 sub-group Ai of order gM. We may now repeat the same 
 
 C 
 reasoning with -j- ; and in this way we must at last reach 
 
 a self-conjugate sub-group of G of order m. Hence, since G 
 contains self-conjugate sub-groups of orders p"" and m, which 
 are relatively prime, G must be the direct product of these 
 sub-groups. 
 
 It should be noticed that the above reasoning does not 
 necessarily hold if G is not soluble ; for then the order 
 of H' may be of the form p'^f^fi, where fi, the order of a simple 
 group, contains more than one prime factor. In that case it 
 would not be necessarily true that R' contains a group of order 
 
 101. Though the actual determination of all types of soluble 
 groups of a given order more properly forms part of the subject of 
 Chapter XI, we will here, as a further illustration of the subject of 
 the present Chapter, deal with the problem for groups whose orders 
 are of the form p'q, p and q being distinct primes. 
 
 Every such group must be soluble. In fact, if ^ > 5', the group 
 must contain a self -con jugate sub-group of order p^ ; and if jo < q, there 
 must be a self- conjugate sub-group of order q unless p = 2, q = S; 
 while in this case if there are 4 sub-groups of order 3, there must be 
 a self-conjugate sub-group of order 4. These statements all follow 
 immediately from Sylow's theorem. 
 
 The Abelian groups of order p^q may be specified immediately ; 
 and therefore in what follows we will assume that the group is not 
 Abelian. There are 3 possible arrangements for the composition- 
 factors, viz. 
 
 P) p, 9; p, g, p; q, p, p- 
 
101] SPECIAL TYPES OF GROUP 133 
 
 If the factors are capable of all three arrangements, the group is 
 the direct product of groups of orders jy^ and q; it is therefore an 
 Abelian group. 
 
 If the two arrangements p, j), q and q, p, p are possible, there are 
 self- conjugate sub-groups of orders p^ and q; the group again is 
 Abelian, and all three arrangements are possible. 
 
 There are now five other possibilities. 
 
 I. p, p, q and p, q, p, the only possible arrangements. 
 
 There must be here a sub-group of order pq, containing self- 
 conjugate sub-groups of orders p and q and therefore Abelian. Let 
 this be generated by operations P and Q, of orders p and q. Since the 
 group has sub-groups of order p^, there must be operations of orders 
 p or p"", not contained in the sub-group of order py, and permutable 
 with P. Let R be such an operation, so that R'' belongs to the 
 sub-group of order pq. R cannot be permutable with Q, as the 
 group would be then Abelian ; hence 
 
 R-^QR = Q\ 
 
 so that R-'-QRP = Q"', 
 
 and a^ = 1 (mod. q). 
 
 This case can therefore only occur if ^ is a factor of q — l. There are 
 two distinct types, according as R is of order p or of order p^ ; i.e. 
 according as the sub-groups of order p- are non-cyclical or cyclical. 
 If a and /3 are any two distinct primitive roots of the congruence 
 
 a? = 1 (mod. q), 
 the relations R-^QR = Q% 
 
 and B-'QR=Q^, 
 
 do not lead to distinct types, since the latter may be reduced to the 
 form by replacing Rhj R", where 
 
 fi'^a (mod. q). 
 The two types are respectively defined by the relations 
 $«=1, PP=1, RP=l, I-'QP=Q, 
 R-^PR = P, R-^QR=Q''; 
 and Q^ = l, R"'^!, R-'QR^Q". 
 
 In each case, a is a primitive root of the congruence aP = 1 (mod. q). 
 
 II. p, q, p and q, p, p, the only possible arrangements. 
 
 There must be a self-conjugate sub-group of order pq, in which 
 the sub-group of order q is not self -conjugate, and a self -conjugate 
 sub-group of order p^. The sub-group of order pq must be given by 
 
 a« = 1 (mod. p) ; 
 
134 SPECIAL TYPES [101 
 
 SO that in this case q must be a factor of ^ - 1. If the sub-group of 
 order p^ is not cyclical, there must be an operation R of order p, not 
 contained in the sub-group of order pq. Any such operation must 
 be permutable with P. Moreover since the sub-group of ovAqv pq is 
 self-conjugate and contains only p sub-groups of order q, the 
 sub-group {Q] must be permutable with some operation of order p. 
 Hence we may assume that R is permutable with [Q], and, since 
 p> q, with Q. 
 
 We thus obtain a single type defined by 
 
 Pi'=l, RP=l, PR = RP, 
 
 Q^ = l, QR = RQ, Q-'PQ = P^. 
 
 If the sub-group of order p^ is cyclical, all the operations, which 
 have powers of p for their orders and are not contained in the sub- 
 group of order pq, must be of order p^. There can therefore be no 
 operation of order p, which is permutable with {Q} ; therefore there 
 is no corresponding type. 
 
 III. p, p, q, the only possible arrangement. 
 
 There must be a self -conjugate sub-group of order 2}q, which has 
 no self -conjugate sub-group of order p ; it is therefore defined by 
 
 i^=l, e«=l, P-^QP=Q\ 
 a* = 1 (mod. q) ; 
 so that here pi must be a factor of q—\. 
 
 If the sub-groups of order p' are not cyclical, there must be an 
 operation R' of order p, not contained in this sub-group and per- 
 mutable with P. Hence 
 
 R'-^QR'=Q^; 
 
 and if fi^a^ (mod. q), 
 
 then R'P'" is an operation of order p, which is not contained in the 
 sub-group of order pq and is permutable with Q. It is therefore a 
 self-conjugate operation of order p. Hence /), q, j) is a possible 
 arrangement of the composition-factors, and there is in this case 
 no type. 
 
 If the sub-groups of order p^ are cyclical, there must be an opera- 
 tion R "of order p^, such that 
 
 Rf = P. 
 
 Hence R-'QR = Q^, 
 
 where (3 is a, primitive root of the congruence 
 
 IP'=1 (mod. g). 
 
 This case then can only occur when p'' is a factor of q—i', and we 
 again have a single type defined by 
 
 RP' = 1, Q^ = 1, R-^QR = Q^ 
 
101] OF GROUP 135 
 
 IV. p, q, p, the only possible arrangement. 
 
 Here the self-conjugate sub-group of order pq must be given by 
 
 PP = \, Qi = \, Q-^PQ = P% 
 
 a« = 1 (mod. p), 
 
 and q must be a factor of p — \. As in II, there must be an 
 operation R of order p, permutable with {Q] and therefore with Q ; 
 and since B transforms {P, Q} into itself, it must be permutable 
 with P. This however makes the sub-group {P, R} self -conjugate, 
 which requires q, p, p to be a possible arrangement of the composi- 
 tion-factors. Hence there is no type corresponding to this case. 
 
 V. q, p, pi, the only possible arrangement. 
 
 If the sub-group of order p'^ is cyclical, and is generated by P, 
 while Q is an operation of order q, we must have 
 
 Q-'PQ=P^, 
 
 where a* h 1 (mod. p'^). 
 
 Here q must be a factor of ^— 1 ; since the congruence has just 
 q — \ primitive roots, there is a single type of group. 
 
 If the sub-group of order p'^ is not cyclical, it can be generated 
 by two permutable operations P^ and P„ of order p. Since a 
 sub-group of order q is not self-conjugate, either p or pi^ must be 
 congruent to unity (mod. q) ; and therefore q must be a factor of 
 either p-\ or ^ -i- 1. 
 
 Suppose iirst, that g is a factor of ^j — 1. 
 
 There are JJ + 1 sub-groups of order ^;. When these are 
 transformed by any operation Q of order q, those which are not 
 permutable with Q must be interchanged in sets of q. Hence at 
 least two of these subgroups must be permutable with Q, and we 
 may take the generating operations of two such sub-groups for P^ 
 and P^. Therefore 
 
 Q-'P,Q = P,% Q''P,Q = Pi. 
 
 Now if either a or ^8, say j8, were unity, then [Q, P-^] would be a 
 self -conjugate sub-group and jo, q, p would be a possible arrangement 
 of the composition-factors. Hence neither a nor /J can be unity, and 
 we may take 
 
 where a is a primitive root of 
 
 a^=\ (mod. p), 
 and X is not zero. 
 
 It remains to determine how many distinct types these equations 
 contain. When y = 2, the only possible value of x is unity ; and there 
 is a single type. When q is an odd prime, and we take 
 Qy = Q', P, = P,', P, = A', xy = 1 (mod. q), 
 
136 SPECIAL TYPES [101 
 
 the equations become 
 
 Q'-'P,'Q' = Pr, Q'-'P^Q' = Pf, 
 and therefore the values x and y of the index of a, where 
 xy=\ (mod. q), 
 
 give the same type. Now the only way, in which the two equations 
 can be altered into two equations of the same form, is by replacing 
 Q by some other operation of the group whose order is q and by 
 either interchanging Pi and P^ or leaving each of them unchanged. 
 Moreover the other operations of the group whose orders are q are 
 those of the form Q'-P-^P^, where I is not zero, and this operation 
 transforms Pj and P^ in the same way as ()'. Hence finally, the 
 values X and y of the index will only give groups of the same type 
 when 
 
 xy=\ (mod. q). 
 
 There are therefore J (g + 1) distinct types, when q is an odd prime; 
 they are given by the above equations. 
 
 Suppose next, that g is a factor oi p+\. 
 
 Any operation Q, of order q, will transform the sub-groups of 
 order p, with which it is not permutable, so as to interchange them 
 in sets of q ; and hence if it is permutable with any sub-group, it 
 must be permutable with q at least. This, by the last case, is 
 clearly impossible, and hence Q is permutable with no sub-group of 
 order p. We may therefore, by suitably choosing the generating 
 operations of the group of order p^, assume that 
 
 Q-'P,Q-=P„ Q-^P,Q = P:P,^ 
 
 If now Q^^'-'P^Q"^' = Pi-^P/^, 
 
 then a^+i = a^^, /3j,+i E a^ + 13^^ (mod. p), 
 
 and therefore /3,.+i - (i^^ - a/S^j-i E (mod. p). 
 
 Hence if ti and i^ are the roots of the congruence 
 
 i^-j3L-a=0 (mod. ja), 
 
 , a;+l _ , 03+1 
 
 then ^a;= — - 
 
 Now since Q'^ is the lowest power of Q that is permutable with 
 Pj, ^^_i must be the first term of the series /3i, /Sj, ... which vanishes. 
 Hence q is the least value of z for which 
 
 h = 'i J 
 and therefore the congruence 
 
 i^ - ySi - a E 
 
 is irreducible. Moreover aj_i must be congruent to unity, and 
 therefore 
 
 1 = - t,i, 2 !!_ = tj«. 
 
101] OF GROUP 137 
 
 From the quadratic congruence satisfied by i, it follows that 
 aE-iP+^s-l, /3=iP+t (mod. ^j); 
 
 and thence a^ = ^ , ^^ = . 
 
 Finally, we may shew that, when y is a factor of p+l, the 
 equations 
 
 P/=l, Pi=\, Q'i=\, F,P, = P„F„ 
 
 Q-'P,Q = P, , Q-U\Q = Pr'P/^\ 
 
 where i is a primitive root of the congruence 
 
 t' = 1 (mod. p), 
 
 define a single type of group, whatever primitive root of the con- 
 gruence is taken for t. 
 
 Thus from the given equations it follows that 
 q-P,Qr = Pr-'Pi'-' = P,, say, 
 and Q-^P^Q' = (p^^-^p/'-y-^ {P,'^Pff'-\ 
 
 p eLaPx—l—o^x—lPx p Oa— l+)3a; 
 
 — •* 1 -^ 3 
 
 _ p -1 p ^'+1.' 
 
 — -^ 1 -^ 3 
 
 If then we take P^ P^ and Q" as generating operations in the place 
 of Pi, P^ and Q, the defining relations are reproduced with i"' in the 
 place of I. The relations therefore define a single type of group '. 
 
 We have, for the sake of brevity, in each case omitted the 
 verification that the defining relations actually give a group of 
 order p^q. This presents no difiiculty, even for the last type ; for 
 the previous types it is immediately obvious. 
 
 1 On groups whose order is of the form p^q the reader may consult ; Holder, 
 "Die Gruppeu der Ordnungen p^, pq^, pqr, p*,'' Math. Ann. xliii (1893), in 
 particular pp. 335 — 360; and Cole and Glover, "On groups whose orders are 
 products of three prime factors," Amer. Journal, xv (1893), pp. 202 — 214. 
 
CHAPTER YIII. 
 
 ON SUBSTITUTION- GROUPS : TRANSITIVE AND 
 INTRANSITIVE GROUPS. 
 
 102. It has been proved, in the theorem in § 20, that every 
 group is capable of being represented as a group of substitutions 
 performed on a number of symbols equal to the order of the 
 group. For applications to Algebra, and in particular to the 
 Theory of Equations, the presentation of a group as a group of 
 substitutions is of special importance ; and we shall now proceed 
 to consider the more important properties of this special mode 
 of representing groups ^ 
 
 Definition. When a group is represented by means of 
 substitutions performed on a finite set of n distinct symbols, 
 the integer n is called the degree of the group. 
 
 It is obvious, by a consideration of simple cases, that a 
 group can always be represented in different forms as a group 
 of substitutions, the number of symbols which are permuted 
 in two forms not being necessarily the same ; examples have 
 already been given in Chapter II. The " degree of a group " 
 is therefore only an abbreviation of " the degree of a special 
 representation of the group as a substitution-group." 
 
 The n\ substitutions, including the identical substitution, 
 that can be performed upon n distinct symbols, clearly form a 
 
 1 When it is necessary to call attention directly to the fact that the group 
 we are dealing with is supposed to be presented as a group of Substitutions, the 
 group will be spoken of as a substitution-group. 
 
103] SYMMETRIC AND ALTERNATING GROUPS 139 
 
 group ; for they satisfy the conditions of the definition (§ 12). 
 Moreover they form the greatest group of substitutions that can 
 be performed on the n symbols, because every possible substitu- 
 tion occurs among them. When a group then is spoken of as of 
 degree n, it is implicitly being regarded as a sub-group of this 
 most general group of order n ! which can be represented by 
 substitutions of the n symbols; and therefore (Theorem I, 
 § 22) the order of a substitution-group of degree n must be 
 a factor of w !. 
 
 103. It has been seen in § 11 that any substitution per- 
 formed on n symbols can be represented in various ways as 
 the product of transpositions ; but that the number of transpo- 
 sitions entering in any such representation of the substitution 
 is either always even or always odd. In particular, the identical 
 substitution can only be rejjresented by an even number of 
 transpositions. Hence if S and S' are any two even (§11) 
 substitutions of n symbols, and T any substitution at all of n 
 symbols, then SS' and T^^ST are even substitutions. The 
 even substitutions therefore foi'm a self-conjugate sub-group 
 H of the group G of all substitutions. 
 
 If now T is any odd substitution, the set of substitutions 
 TH are all odd and all distinct. Moreover they give all the 
 odd substitutions; for if T' is any odd substitution distinct 
 from T, then T~^T' is an even substitution and must be 
 contained in H. Hence the number of even substitutions is 
 equal to the number of odd substitutions : and the order of G 
 is twice that of H. 
 
 Definitions. The group of order n \ which consists of all 
 the substitutions that can be performed on n symbols is called 
 the symmetric group of degree n. 
 
 The group of order |re' which consists of all the even 
 substitutions of n symbols is called the alternating* group of 
 degree n. 
 
 * The symmetric group has been so called because the only functions of the 
 n symbols which are unaltered by all the substitutions of the group are the 
 symmetric functions. 
 
 All the substitutions of the alternating group leave the squai-e root of the 
 discriminant unaltered (§11). 
 
140 TRANSITIVE [103 
 
 If the substitutions of a group of degree n are not all even, 
 the preceding reasoning may be repeated to shew that its even 
 substitutions form a self-conjugate sub-group whose order is 
 half the order of the group ; and this sub-group is a sub-group 
 of the alternating group of the n symbols. 
 
 104. Definition. A substitution-group is called transitive 
 when, by means of its substitutions, a given symbol a^ can be 
 
 changed into every other symbol a^, a^, , «„ operated oa 
 
 by the group. When it has not this property, the group is 
 called intransitive. 
 
 A transitive group contains substitutions changing any one 
 symbol into any other. For if S and T respectively change Oi 
 into a^ and at, then 8~^T changes a^ into a^ 
 
 Theorem I. The substitutions of a transitive group G, which 
 leave a given symbol a^ unchanged, form a sub-group; and the 
 number of substitutions, which change a^ into any other symbol a„ 
 is equal to the order of this sub-group. 
 
 The substitutions which leave a^ unchanged must form a 
 sub-group H oi 0; for if 8 and S' both leave ai unchanged, so 
 also dt)es S8'. 
 
 Let the operations of G be divided into the sets 
 H, HSi, HS2, , HSm-i- 
 
 No operation of the set HSp leaves a^ unchanged ; and each 
 operation of the set HSp changes a^ into a^, if Sp does so. If 
 the operations of any other set HSq also changed a^ into ap, then 
 8pSq~^ would leave Oj unchanged and would belong to H, which 
 it does not. Hence each set changes ai into a distinct symbol. 
 The number of sets must therefore be equal to the number of 
 symbols, while from their formation each set contains the same 
 number of substitutions. If N is the order and n the degree 
 
 of the transitive group G, then — is the order of the sub- 
 
 N 
 group that leaves any symbol ai unchanged ; and there are — 
 
 substitutions changing a^ into any other given symbol ap. 
 
 Corollary. The order of a transitive group must be 
 divisible by its degree. 
 
105] GROUPS 141 
 
 Every group conjugate to H leaves one symbol unchanged. 
 For if 8 changes a^ into Op, then S'^HS leaves Up unchanged. 
 The sub-groups which leave the different symbols unchanged 
 form therefore a conjugate set. 
 
 A transitive group of degree n and order mn has, as we 
 have just seen, m — 1 substitutions other than identity which 
 leave a given symbol Oj unchanged. Hence there must be at 
 least mn — 1 — ?i(m — 1),- i.e. w — 1, substitutions in the group 
 that displace every symbol. If the m — 1 substitutions, other 
 than identity, that leave a^ unchanged are all distinct from the 
 m — 1 that leave ap unchanged, whatever other symbol ap may 
 be, n—\ will be the actual number of substitutions that 
 displace all the symbols ; and no operation other than identity 
 will displace less than n — \ symbols. If however the sub- 
 gi-oups that leave a^ and Op unchanged have other substitutions 
 besides identity in common, these substitutions must displace 
 less than w — 1 symbols ; and there will be more than n—1 
 substitutions which displace all the symbols. 
 
 Ex. If the substitutions of two transitive groups of degree n 
 which displace all the symbols are the same, the groups can only 
 differ in the substitutions that keep just one symbol unchanged. 
 
 (Netto.) 
 
 105. We have seen that every group can be represented as a 
 substitution-group whose degree is equal to its order. A reference 
 to the proof of this theorem (§ 20) will shew that such a substitution- 
 group is transitive, and that the identical substitution is the only 
 one which leaves any symbol unchanged. 
 
 We wiD now consider some of the properties of a transitive 
 group of degree n, whose operations, except identity, displace all or 
 all but one of the n symbols. It has just been seen that such a group 
 has exactly n-\ operations which displace all the n symbols. 
 If these n — \ operations, with identity, form a sub-group, the sub- 
 group must clearly be self- conjugate. 
 
 Suppose now that nm is the order of the group. Then the order of 
 the sub-group, that leaves one symbol Oj unchanged, is m ; and if a^ 
 is any other of the n symbols, no two operations of this sub-group 
 can change a^ into the same symbol. For if S, S' were two opera- 
 tions of the sub-group both changing a^ into a^, then SS'~'^ would be 
 an operation, distinct from identity, which would keep both Oj and 
 02 unchanged. Let now P be any operation that displaces all the 
 symbols. Then the set of in operations S~'^PS, where for S is put in 
 
142 SPECIAL FORMS 
 
 [105 
 
 turn eacli operation of the sub-group that keeps a^ unchanged, are all 
 distinct ; for each of them changes a^ into a different symbol. If this 
 set does not exhaust the operation conjugate to P, and if P-^ is another 
 such operation, then the set of m operations S'^P-^S are all distinct 
 from each other and from those of the previous set. This process 
 may be continued till the operations conjugate to P are exhausted. 
 The number of operations conjugate to P is therefore a multiple 
 of m. Since P itself does not belong to any one of the n conjugate 
 sub-groups that each keep one symbol fixed, no operation conjugate 
 to P can belong to any of them. Hence each of the km oi)erations 
 of the conjugate set to which P belongs displaces all the symbols. 
 The n - 1 operations that displace all the symbols can therefore be 
 divided into sets, so that the number in each set is a multiple of m- 
 and hence m must be a factor oi n-1. 
 
 Suppose now that p is any prime factor of n, and that js" is the 
 highest power of p which divides n. If P is an operation whose 
 order is a power of p, and if p^ft. is the order of the greatest sub- 
 group h that contains P self-conjugately, then P is one of — 
 
 conjugate operations. Now (§ 87) the sub-group h contains k/i [h <^ 1) 
 operations whose orders are relatively prime to p; and therefore 
 there are kfn operations of the form PQ, where Q is permutable with 
 P and the order of Q is relatively prime to the order of P. If P' 
 is any operation conjugate to P, there are similarly A/u, operations of 
 the form P'Q' ; and (§ 16) no one of these operations can be 
 identical with any one of those of the previous set. The group 
 
 therefore will contain — g- distinct operations, which are conjugate 
 
 to the various operations of the set PQ. Moreover since P displaces 
 all the symbols, each one of these operations must displace all the 
 symbols. 
 
 If then the group has r distinct sets of conjugate operations 
 whose orders are powers of p, the number of operations, whose orders 
 are divisible by p, is equal to 
 
 "=■■ nmhg 
 8=1 P^' ' 
 Also the number of operations, which displace all the symbols 
 
 and the orders of which are not divisible by p, is of the form — ^ - 1 
 
 (§ 87). 
 
 Hence finally 
 
 a^af-i-(!4»-i)} 
 
 s=i p^' n\, V^" J} 
 
105] OF TRANSITIVE GROUPS 143 
 
 The greatest possible value of m will correspond to the suppositions 
 
 r=l, k, = \, k^=l; 
 
 and these give m=p'^—l. 
 
 Hence if n = p'^q^ ... r* , where p, q, ..., r are distinct primes, m 
 cannot be greater than the least of the numbers 
 
 p'^—1, q^-1, ..., r'^'-l. 
 
 Certain particular cases may be specially noticed. First, a group 
 of degree n and order n {n — 1), whose operations other than 
 identity displace all or all but one of the symbols, can exist only 
 when n is the power of a prime'. Groups which satisfy these 
 conditions will be discussed in § 112. 
 
 Similarly, a group of degree n and order nm., where m is not less 
 than Jn, whose operations other than identity displace all or all but 
 one of the symbols, can only exist when n is the power of a prime. 
 
 If n is equal to twice an odd number, a transitive group of 
 degree n, none of whose operations except identity leave two symbols 
 unchanged, must be of order n. 
 
 Lastly we may shew that, if m is even, a group of degree n and 
 order nvi, none of whose operations except identity leave two symbols 
 unchanged, must contain a self -conjugate Abelian sub-group of order 
 and degree n. 
 
 A sub-group that keeps one symbol fixed must, if m is even, 
 contain an operation of order 2. If it contained r such operations, 
 the group would contain nr ; and each of these could be expressed 
 as the product of |(« - 1 ) independent transpositions. Now from n 
 symbols ^n(n-l) transpositions can be formed. If then r were 
 greater than 1, among the operations of order 2 that keep one 
 symbol fixed there would be pairs of operations with a common 
 transposition ; and the product of two such operations would be an 
 operation, distinct from identity, which would keep two symbols at 
 least fixed. This is impossible ; therefore r must be unity. Now let 
 
 be the n operations of order 2 belonging to the group. Since no 
 two of these operations contain a common transposition, 
 
 are the w — 1 operations which displace all the symbols. These 
 operations may also be expressed in the form 
 
 A^Ai, A^A^, , A^Ay^^i, A^A,.^i, , jlj..fl„j 
 
 and since 
 
 ApA p . Ay.Aq = ApAqy 
 
 1 Jordan, " E^cherches sur les substitutions,'' Liouville's Journal, 2™* s^r. 
 Vol. XVII (1872), p. .S55. 
 
144 SELF-CONJUGATE [105 
 
 the product of any two of these operations is either identity or 
 another operation which displaces all the symbols. Hence the n-l 
 operations which displace all the symbols, with identity, form a self- 
 conjugate sub-group. Now 
 
 SO that A^ transforms every operation of this sub-group into its 
 inverse. Hence 
 
 A^Ap . A^A,. = Af^Ap = AqA^ . Aj.A^ ; 
 i.e. every two operations of this sub-group are permutable, and the 
 sub-group is therefore Abelian. 
 
 106. If S = (a^a2 ai)(ai+iai+2 Oj) , 
 
 and j,^fyy ,an\ 
 
 VOi, 62, , bj' 
 
 are any two substitutions of a group, then (§ 10) 
 
 T-^ST={bA bi)(bi+A+. bj) 
 
 Hence every substitution of the group, which is conjugate 
 to 8, is also similar to /S. It does not necessarily or generally 
 follow that two similar substitutions of a group are conjugate. 
 That this is true however of the symmetric group is obvious, 
 for then the substitution T may be chosen so as to replace the 
 n symbols by any permutation of them whatever. 
 
 A self-conjugate substitution of a transitive group of degree 
 n must be a regular substitution (| 9) changing all the n symbols. 
 For if it did not change all the n symbols, it would belong to 
 one of the sub-groups that keep a symbol unchanged. Hence, 
 since it is a self-conjugate substitution, it would belong to each 
 sub-group that keeps a symbol unchanged, which is impossible 
 unless it is the identical substitution. Again if it were not 
 regular, one of its powers would keep two or more symbols 
 unchanged, and this cannot be the case since every power of a 
 self-conjugate substitution must be self-conjugate. On the 
 other hand, a self-conjugate sub-group of a transitive group 
 need not contain any substitution which displaces all the 
 symbols. Thus if 
 
 S = (12)(S4>), 
 
 T = (135) (246), 
 then [S, T} is a transitive group of degree 6. The only 
 substitutions conjugate to S are 
 
 7^'ST={34>)(56) and TST-' = (12) (5Q) ; 
 
107] SUBSTITUTIONS 145 
 
 and these, with S and identity, form a self-conjugate sub-group 
 of order 4, none of whose substitutions displace more than 4 
 symbols. The form of a self-conjugate sub-group of a transitive 
 group will be considered in greater detail in the next Chapter. 
 
 107. Since a self-conjugate substitution of a transitive 
 substitution-group G of degree n must be a regular substitution 
 which displaces all the symbols, the self-conjugate sub-group 
 H oi G which consists of all its self-conjugate operations must 
 have n or some submultiple of n for its order. For if 8 and S' 
 are two self-conjugate substitutions of G, so also is S~^S' ; and 
 therefore 8 and 8' cannot both change a into b. The order of 
 H therefore cannot exceed n ; and if the order is n', the 
 substitutions of H must interchange the n symbols in sets of n', 
 so that n' is a factor of n. Let now >S, some substitution 
 performed on the n symbols of the transitive substitution-group 
 G of degree n, be permutable with every substitution of 
 G. Then >S is a self-conjugate operation of the transitive 
 substitution-group {8, G} of degree n, and it is therefore a 
 regular substitution in all the n symbols. The totality of the 
 substitutions 8, which are permutable with every substitution 
 of G, form a group (not necessarily Abelian); and the order of 
 this group is M or a factor of n. 
 
 The special case, in which G is s, group whose order n is 
 equal to its degree, is of sufficient importance to merit par- 
 ticular attention. If 
 
 8i(= 1), 82, , 8n 
 
 are the operations of G, it has been seen (§ 20) that the 
 substitution-group can be expressed as consisting of the n 
 substitutions 
 
 / Oi , O2 , On 
 
 loo 00 <?<?/' \^ ~ ^> ^> > '^/i 
 
 \OiOx) 020a;, , ^n^x' 
 
 performed upon the symbols of the operations of the group. 
 
 Now if pre-multiplication be used in the place of post- 
 multiplication, it may be verified exactly as in § 20 that the n 
 substitutions 
 
 Oi , O2 , , On 
 
 pa;Oi, 0x02, , OjjOn' 
 
 B. 10 
 
146 SUBSTITUTIONS PERMUTABLE WITH [107 
 
 form a group 0' simply isomorphic with G; a,nd the substitu- 
 tion of 0' which is given corresponds to the operation Sx~^ of G. 
 Moreover 
 
 Si , 8n \ ^ f Si , , Sn \ ^ Si , 8n 
 
 .SxSi, , SxSn' \SiSy, , SnSyJ \SxSi , S-cS„. 
 
 SxSi , SxSn")^ / SiSy , , SnSy \ 
 
 SlSy, , SlSnJ \SxSlSy, , SxSnSyJ 
 
 SxSi , , SxSn \_( Si , , *Si„ \ 
 
 SxSiSy, , SxSnSyJ \SlSy, , SnSyJ' 
 
 SO that every operation of G' is permu table with every opera- 
 tion of G, while G and G' can be expressed as transitive 
 substitution-groups in the same n symbols. Hence : — 
 
 Theorem II. Those substitutions of n symhols which are 
 permutable with every substitution of a substitution-group G of 
 order n, transitive in the n symbols, form a group G' of order 
 and degree n, simply isomorphic with G *. 
 
 It has, in fact, been seen that there is a substitution-group 
 G' of order and degree n, every one of whose substitutions is 
 permutable with every substitution of G ; and also that the 
 order of any such group cannot exceed n. 
 
 If Sx is a self-conjugate substitution of G, the substitutions 
 
 Si , , Sn \ n / Si , , Sn 
 
 SiSx , SnSxJ \SxSi, , SxSn^ 
 
 are the same. Hence G and G' have for their greatest common 
 sub-group, that which is constituted by the self-conjugate 
 substitutions of either ; and if n' is the order of this sub-group, 
 
 the order o{{G,G'}is—. In particular, if G is Abelian, G and 
 
 G' coincide; and if G has no self-conjugate operation except 
 identity, {G, G'} is the direct product of G and G'. 
 
 The sub-group of [G, G'} which leaves one symbol, say 8i, 
 unchanged, is formed of the distinct substitutions of the set 
 
 f Si , S^ , , S„ \ , . 2 n). 
 
 [Sx-'SiSx, Sx~'SA, ,Sx-^s,,sJ' ^^ ''^'- ' ^ 
 
 * Jordan, Traiti des Substitutions (1870), p. 60. 
 
107] EVERY SUBSTITUTION OF A GROUP 147 
 
 When has no self-conjugate operation except identity, 
 the order of this sub-group is n, and it is simply isomorphic 
 with 0. In fact, in this case the order of [0, 0'], a transitive 
 group of degree n, is w^ and therefore the order of a sub-group 
 that keeps one symbol unchanged is n. Again 
 
 "1 ) l^i , , 'Jji 
 
 Ox OiOx) Ox O^Ox, , ^2: OyiOx' 
 
 \Oy~^OlOy, Sy~^S2Sy, , Sy~^SnSy/ 
 
 _ ( Si , S2 , , S„ 
 
 \Sx~'^SiSx, 8x~^82S!c, , 8x~^Sn8, 
 
 Ox OiOx , Ox O^Ox 3 , Ox OnOx 
 
 Oy Ox OjOxOyj Oy Ox O^OxOy, , Oy O/p Oj^O/gOy, 
 
 (Oi , O2 , , On 
 
 8y~^8x~'^Si8x8y, 8y~''^8x~^828xSy, , Sy~'^8x~^8nSx8y 
 
 thus giving a direct verification that the sub-group is isomorphic 
 with the group whose operations are 
 
 Wi) '^2, , On- 
 
 When G contains self-conjugate operations, it will be 
 multiply isomorphic with the sub-group K^ of {G, G'} which 
 keeps the symbol 8^ fixed ; and if ^ is the group constituted by 
 the self-conjugate operations of G (or G'), then K^ is simply 
 
 isomorphic with — . 
 9 
 
 If Ki is not a maximum sub-group of {G, G'}, let / be a 
 
 greater sub-group containing K^ . Then / and G' (or G) must 
 
 contain common substitutions. For every substitution of 
 
 {G, G'} is of the form 
 
 Oi , O2 1 , On 
 
 8y8i8x, 8y 828x1 , 8y8n8i 
 
 and if this substitution belongs to /, then 
 / Si , 82 , , 8n \ 
 
 \8ySlSx, Sy82Sx, > SySnSxJ 
 
 f 81 , 82 , Sn \ 
 
 [Sx8iS^-\ 8x828,-' , 8A8c.-y 
 
 I Si , 82 , Sn \ 
 
 \Sx8y8i, 8x8y82, ,8xSySj' 
 
 10—2 
 
 J^ 
 
 or 
 
148 MULTIPLY TRANSITIVE [107 
 
 is a substitution of G' which belongs to I. Moreover, since G' 
 is a self-conjugate sub-group of [Q, 0'], the substitutions of 0' 
 which belong to I form a self- conjugate sub-group of 7: this 
 sub-group we will call H'. 
 
 Now every substitution of the group can be represented as 
 the product of a substitution of K^ by a substitution of G : and 
 therefore all the sub-groups conjugate to / will be obtained on 
 transforming / by the operations of G. Hence, because every 
 substitution of G transforms H' into itself, H' is common to 
 the complete set of conjugate sub-groups to which / belongs ; 
 and H' is therefore a self-conjugate sub-group of {G, G'}. 
 Finally then, K^ is a maximum sub-group of {G,G'}, if and only 
 if (? is a simple group. 
 
 108. Definition. A substitution-group, that contains one 
 or more substitutions changing k given symbols Oj, Oj, ...,aj; 
 into any other k symbols, is called k-ply transitive. 
 
 Such a group clearly contains substitutions changing any 
 set of k symbols into any other set oi k; and the order of the 
 sub-group keeping any j(:^ k) symbols unchanged is independent 
 of the particular j symbols chosen. 
 
 Theorem III. The order of a k-ply transitive group of 
 
 degree n is n(n — V) {n — k + V)m, where m is the order of 
 
 the sub-group that leaves any k symbols unchanged. This 
 sub-group is contained self-conjugately in a sub-group of order 
 k\m. 
 
 If N is the order of the group, the order of the sub-group 
 
 N 
 which keeps one symbol fixed is — , by Theorem I (§ 104). 
 
 Now this sub-group is a transitive group of degree n — 1; and 
 
 therefore the order of the sub-group that keeps two symbols 
 
 N 
 unchanged is —z rr . If A; > 2, this sub-group again is a 
 
 transitive group of degree w — 2 ; and so on. Proceeding thus, 
 the order of the sub-group which keeps k symbols unchanged 
 is seen to be 
 
 N 
 
 nin-1) (n-k+l)' 
 
 which proves the first part of the theorem. 
 
108] GROUPS 149 
 
 Let Oi, ttj, , aj. be the k symbols which are left un- 
 changed by a sub-group H of order m. Since the group is 
 ^-ply transitive, it must contain substitutions of the form 
 
 ., aj., &, c, 
 
 /«!, (h, , Ifc, b, c, \ 
 
 W. <, , dk, h',c', /' 
 
 where a/, cij', ,%' are the same k symbols as aj, a^, , a^ 
 
 arranged in any other sequence. Also every substitution of this 
 form is permutable with H, since it interchanges among them- 
 selves the symbols left unchanged by H. Further, if Si and 
 82 are any two substitutions of this form, S^'^^S^ will belong to 
 H if, and only if, S^ and S^ give the same permutation of the 
 
 symbols Oj, a^, , aj.. Hence finally, since k\ distinct 
 
 substitutions can be performed on the k symbols, the order of 
 the sub-group that contains H self-conjugately is k\m. 
 
 If m is unity, the identical substitution is the only one that 
 keeps any k symbols fixed, and there is just one substitution 
 that changes k syaibols into any other k. In the same case, the 
 group contains substitutions which displace n — k + 1 symbols 
 only, and there are none, except the identical substitution, 
 which displace less. 
 
 If m > 1, the group will contain m— 1 substitutions besides 
 identity, which leave unchanged any k given symbols, and 
 therefore displace n — k symbols at most. 
 
 It follows from § 105 that a i-ply transitive group of degree 
 
 n and order n(n — l) (n — k + 1) can exist only i{ n — k+2 
 
 is the power of a prime. For such a group must contain 
 sub-groups of order (n — k + 2)(n- k + 1), which keep k—2 
 symbols unchanged and are doubly transitive in the remaining 
 n — k + 2. When k is n, the group is the symmetric group ; and 
 when k is n - 2, we shall see (in § 110) that the group is the 
 alternating group. If k is less than n— 2, M. Jordan^ has 
 shewn that, with two exceptions for m = 11 and n=12, the 
 value of k cannot exceed 3. The actual existence of triply 
 transitive groups of order (p"-|-l)^"(p"— 1), for all prime 
 values of p, will be established in § 113. 
 
 ' "E&herohes surles substitutions," Lioiiville's Journal, 2"= s6r., Vol. xvii 
 (1872), pp. 357—363. 
 
150 MULTIPLY TRANSITIVE [109 
 
 109. A k-Tply transitive group, of degree n and order 
 N, is not generally contained in some (A; + l)-ply transitive 
 group of degree n + 1 and order iV(re + l). To determine 
 whether this is the case for any given group, M. Jordan' has 
 suggested the following tentative process, which for moderate 
 values of n is always practicable. 
 
 Let G* be a transitive group of order N in the n symbols 
 
 (h> 0,it ) (^n > 
 
 and suppose that Q is that sub-group of the transitive group F 
 in the n + \ symbols 
 
 which leaves the symbol a,i+i unchanged. Then F must be at 
 least doubly transitive, and it therefore contains a substitution 
 of order 2 which interchanges the two symbols ck and a„+i. 
 Let A be such a substitution. Then 
 
 for {0, A} is contained in F, and its order cannot be less than 
 the order of F. Also if S is any substitution of G, two other 
 substitutions S' and 8" of G can always be found such that 
 
 ASA = S'AS". 
 
 In fact, if ASA changes a^ into an+i, and if S' changes a, 
 into tti, then AS'~^ASA leaves a^+i unchanged, and it therefore 
 belongs to G. 
 
 Conversely, if A is any operation of order 2 which changes 
 Oj into ttn+i and permutes the remaining a's among themselves ; 
 and if, whatever substitution of G is represented by S, two other 
 substitutions of G can be found such that 
 
 ASA = S'AS"; 
 
 then {(?, -4} is a group with the required properties. In fact, 
 when these conditions are satisfied, every substitution of the 
 group {G, A] can be expressed in one of the two forms 
 
 Si, S2AS3, 
 
 1 TraiU dee Substitutiom (1870), pp. 31, 32. 
 
110] GROUPS 151 
 
 where Si, 82, 83 are substitutions of Q. For instance, the 
 substitution 
 
 A8pASgA8r = 8p'A8p". 8qA8r, 
 
 = 8p'AStASr, a Sp"S, = St, 
 
 = 8p'St'ASt"8r, 
 which is of the second form. 
 
 Moreover every operation of the form S^ASa displaces ««+!> 
 and therefore the sub-group of {0, A] which leaves a^+i un- 
 changed is G. 
 
 It is clearly sufficient that the conditions 
 ASA = 8' AS" 
 
 should be satisfied, when each of a set of independent generating 
 operations of G* is taken for S. 
 
 Ex. Construct a doubly transitive group of degree 12 of which 
 the sub-group that keeps one symbol unchanged is 
 
 {(123456789a6), (256a4) (39687)}. 
 
 110. Let 
 
 S={aia2 ai) ( aj_ia^{aj+i (h-iak ) 
 
 be a substitution of a A;-ply transitive group displacing s(>k) 
 symbols. If _; < A; — 1, take 
 
 yOi, a<i, , %_!, Oj;, — 
 
 where \ is some other symbol occurring in 8. Since the 
 group is i-ply transitive, it must contain a substitution such as 
 T. Now 
 
 T-^8T=(cha^ ai) ( aj_iaj)(aj+i a^-ih ) ; 
 
 and this is certainly not identical wfth 8, so that T-^ST8~^ 
 
 cannot be the identical substitution. Moreover aj, a^, ,cik-i 
 
 are not affected by T~'^8TS~'-; and therefore this substitution 
 will displace at most 2s — 2k + 2 symbols. 
 
 li j = k—l, take 
 
 \(2i, 0,2, , djfc— 1, Cj., 
 
 where Cj is a symbol that does not occur in >Si. Then 
 
 T-^8T = (a,a2 aj) ( ajt-aaj^O {c„ ), 
 
152 MULTIPLY TRANSITIVE [HO 
 
 and this cannot coincide with *S. Now in this case, Oi.cij, ,aj._i 
 
 are not affected by T~^STS''^ ; and therefore this operation 
 will again displace at most 2s— 2k + 2 symbols. 
 
 If then 2s-2k+2<s, 
 
 or s<2k — 2, 
 
 the group must contain a substitution affecting fewer symbols 
 than S. This process may be repeated till we arrive at a 
 substitution 
 
 2 = (ai«2 ai)(ai+i a?) («)+i «*). 
 
 which affects exactly k symbols ; and if this substitution be 
 transformed by 
 
 rp _ (^1> '^•2< > *J:-l) C'fcj 
 
 then 
 
 T-'1,T = {a,a^ «i)(ai+i a,) (oj+i a^-A), 
 
 and S-7-^22'=(a,/3,«^+0- 
 
 Thus in the case under consideration the group contains 
 one, and therefore every, circular substitution of three symbols; 
 and hence (§ 11) it must contain every even substitution. It 
 is therefore either the alternating or the symmetric group. If 
 then a ^-ply transitive group of degree n does not contain the 
 alternating group of n symbols, no one of its substitutions, 
 except identity, must displace fewer than 2k — 2 symbols. 
 It has been shewn that such a group contains substitutions 
 displacing not more than n — k + 1 symbols ; and therefore, for 
 a k-Tp\y transitive group of degree n, other than the alternating 
 or the symmetric group, the inequality 
 n-k+li2k-2, 
 or ' k1f^n+l, 
 
 must hold. Hence : — 
 
 Theorem IV. A group of degree n, which does not con- 
 tain the alternating group of n symbols, cannot he more than 
 {\n + l)-ply transitive. 
 
 The symmetric group is n-ply transitive ; and, since of the 
 two substitutions 
 
 Ki, (^2, ,an—2!(l'ti—i>0'n\ J ((htOjit , Oin—it ^n—\> '^n 
 
 bi, O2, , 611—2, "n—i, bn' \bi, 62, , On— 2, On, On-iJ 
 
Ill] GROUPS 153 
 
 one is evidently even and the other odd, the alternating group 
 is {n — 2)-ply transitive. The discussion just given shews that 
 no other group of degree n can be more than (|n + l)-ply 
 transitive \ 
 
 111. The process used in the preceding paragraph may be 
 applied to shew that, unless w = 4, the alternating group of n 
 symbols is simple. It has just been shewn that the alternating 
 group is (n — 2)-ply transitive. Therefore, if /S is a substitution 
 of the alternating group displacing fewer than n — 1 symbols, a 
 substitution T~^ST can certainly be found such that S~^T~^ST 
 is a circular substitution of three symbols. In this case, the 
 self-conjugate group generated by S and its conjugate substi- 
 tutions contains all the circular substitutions of three symbols, 
 and therefore it coincides with the alternating group itself. If 
 8 displaces n—1 symbols, then T~^ST can be taken so that 
 S-'T-'ST displaces not more than 2(n-l)-2(n-2) + 2, or 
 4 symbols ; and if *S displaces n symbols, 8~'^T~^ST can be found 
 to displace not more than 2n — 2(n — 2) + 2, or 6 symbols. 
 
 It is therefore only necessary to consider the case n=5, when 
 8 displaces n — 1 symbols ; and the cases n = 4, 5, 6, when 8 dis- 
 places n symbols; in all other cases, the group generated by 
 8 and its conjugate substitutions must contain circular substi- 
 tutions of 3 symbols. 
 
 When n = 5, and 8 is an even substitution displacing 4 
 symbols, we may take 
 
 , 
 
 5 = (12) (34). 
 
 If 
 
 r=(12)(35). 
 
 then 
 
 T-^8T= (12) {4<5), 
 
 and 
 
 8-^T-'8T = (S4>o). 
 
 Hence, in this case again, we are led to the alternating group 
 itself 
 
 ' For a further discussion of the limits of transitivity of a substitution- 
 group,] compare, Jordan, Traite des Substitutions, pp. 76 — 87 ; and Boohert, 
 Math. Ann., xxix, (1886) pp. 27—49 ; xxxiii, (1888) pp. 573—583. 
 
154 THE ALTERNATING GROUP IS SIMPLE [lH 
 
 When w = 6, and S is an even substitution displacing all 
 the symbols, we may take 
 
 /S= (12) (3456), 
 or 8' = (123) (456). 
 
 If now T= (12) (3645), 
 
 then S-'T-'ST = {356), 
 
 and <S'-iT-i;ST = (14263); 
 
 and, in either case, we are led to the alternating group. 
 
 When n = 5, and S is an even substitution displacing all 
 the symbols, we may put 
 
 /Sf= (12345). 
 
 If r = (34.5), 
 
 then S-^T-'8T=(1S4!); 
 
 and again the alternating group is generated. 
 
 When K = 4, and 8 is an even substitution displacing all the 
 symbols, we may take 
 
 <S = (12)(34). 
 
 Here the only two substitutions conjugate to 8 are clearly 
 (13) (24) and (14) (23), which are permutable with each other 
 and with 8. Hence the alternating group of 4 symbols, which 
 is of order 12, has a self-conjugate sub-group of order 4. 
 
 Finally when n = 3, the alternating group, being the group 
 {(123)}, is a simple cyclical group of order 3. Hence : — 
 
 Theorem V. The alternating group of n symbols is a simple 
 group except when n = 4>. 
 
 112. It has been seen in § 108 that the order of a doubly transi- 
 tive group of degree n is equal to or is a multiple oi n{n-\). If it 
 is equal to this number, every substitution of the group, except 
 identity, must displace either all or all but one of the symbols ; for 
 a sub-group of order n—\ which keeps one symbol fixed is transi- 
 tive in the remaining re — 1 symbols, and therefore all its substitutions, 
 except identity, displace all the n-\ symbols. 
 
 Now it has been shewn in § 105 that a transitive group of degree 
 n and order n{n — 1), whose operations displace all or all but one of 
 the symbols can exist only if n is the power of a prime p. The 
 n-\ operations displacing all the symbols are the only operations 
 of the group whose orders are powers of p ; and therefore with 
 
112] SPECIAL TYPES OF TRANSITIVE GROUPS 155 
 
 identity they form a self-conjugate sub-group of order n. Moreover 
 it also follows from § 105 that the n — \ operations of this sub-group 
 other than identity form a single conjugate set. Hence this sub- 
 group must be Abelian, and all its operations are of order p. 
 
 Suppose first that w is a prime p, and that P is any operation of 
 the group of order p. If a is a primitive root of p, there must be 
 an operation S in the group such that 
 
 then S^~'^ is the lowest power of aS' which is permutable with P. 
 Now S must belong to a sub-group of order p — \ that keeps one 
 symbol fixed, and we have just seen that the order of [S] is not less 
 than p -\. The sub-group of order p - 1 is therefore cyclical, and 
 
 <S*-i = 1. 
 
 Hence the group, if it exists, must be defined by 
 
 i^ = l, ,S'^-i=l, S-^PS = P\ 
 
 It is an immediate result of a theorem, which will be proved in the 
 next chapter (§ 123), that this group can be actually represented as 
 a transitive substitution-group of degree p ; this may be also verified 
 directly as follows. 
 
 Let P = (a^a^ . . . ap), 
 
 so that P» = (flSid^+iaj^+i . . . a(^_i)a 
 
 lla+iyj 
 
 where the suffixes are to be reduced (mod. p) ; and suppose that S 
 is a substitution that keeps Oj unchanged. Then since 
 
 S-'PS=P^, 
 
 S must change a.^ into fta+i, «3 into osan+i, and generally, a^. into 
 «(r-i)a+i- Hence 
 
 (S = (a2a„+ia„2+i ...)...; 
 
 and since a is a primitive root of p, there is only a single cycle ; so 
 that 
 
 S = {a^a+ld'a^i-l ■ ■ ■ WaP-^+l)- 
 
 The substitutions P and S thus constructed actually generate a 
 doubly transitive substitution-group of degree p and order p (p — I). 
 
 Without making a complete investigation of the case in which rt 
 is the power of a prime, we go on to shew that, p being any prime, 
 there is always a doubly transitive group of degree p"' and order 
 2f(p^-l), in which a sub-group of order jo™- 1 is cyclical'. 
 
 1 Ou the subject of this and the following paragraph, the reader should 
 consult the memoirs by Mathieu, Liouville's Journal, 2""= Ser., t. v (1860), 
 pp. 9 — 42 ; ib. t. vi (1861), pp. 241 — 323 ; where the groups here considered were 
 first shewn to exist. 
 
156 SPECIAL TYPES [112 
 
 Let i be a primitive root of the congruence 
 i""-' = 1 (mod. p), 
 so that the distinct roots of the congruence are 
 
 i,i',i', ,^''"•-^ 
 
 Every rational function of i with real integral coefficients satisfies 
 the same congruence; and therefore every such function is con- 
 gruent (mod. p) to some power of i not exceeding the {p™- l)th. 
 
 Consider now a set of transformations of the form 
 x' = ax + p (mod. p), 
 
 where a is a power of i, and /3 is either a power of i or zero. Two 
 such transformations, performed successively, give another trans- 
 formation of the same form ; and since a cannot be zero, the inverse 
 of each transformation is another definite transformation; so that 
 the totality of transformations of this form constitute a group. 
 Moreover 
 
 x'=ax + ^ 
 and x' = a'x + li',^'^°^-P^' 
 
 are not the same transformation unless 
 
 a = a' and fi = ^' (mod. p). 
 Hence, since a can take p'^-l distinct values and j3 can take p"" 
 distinct values, the order of the group, formed of the totality of 
 these transformations, is p™ (p™ — 1 ). 
 
 The transformations for which a is unity clearly form a sub- 
 group. If S and T represent 
 
 x' = ax + /i and x' = x + y 
 
 respectively, S'^TS represents 
 
 x' = X + ay. 
 
 Hence the transformations for which u, is unity form a self-conjugate 
 sub-group whose order is p"^. Every two transformations of this 
 sub-group are clearly permutable; and the order of each of them 
 except identity is p. 
 
 Again, the transformations for which /3 is zero form a sub-group. 
 Since every one of them is a power of the transformation i 
 
 this sub-group is a cyclical sub-group of order p"^—l. If the trans- 
 formation just written be denoted by /, then S'^IS is 
 
 x' = ix + ft(l-i). 
 
 Hence the only operations permutable with {/} are its own operations, 
 and therefore {/} is one of jo™ conjugate sub-groups. 
 
112] OF TRANSITIVE GROUPS 157 
 
 The set of transformations 
 
 x' = ax + ^ 
 
 therefore forms a group of order p"^ (p™ - I). This group contains a 
 self -conjugate Abelian sub-group of order p"^ and type (1, 1, ..., 1), 
 and p™ conjugate cyclical sub-groups of order ^'" — 1 , none of whose 
 operations are permutable with any of the operations of the self- 
 conjugate sub-group. 
 Now if the operation 
 
 x' = ax + /3 
 be performed on each term of the series 
 
 0, i, ^^ ,^'"-\ 
 
 it will, since every rational integral function of i with real integral 
 coefficients is congruent (mod. p) to some power of i, change the 
 term into another of the same series ; and since the congruence 
 
 ai"' -(- /3 = aiy + /3 
 gives ^ = 2/ (mod. p"^~^), 
 
 no two terms of the series can thus be transformed into the same 
 term. Moreover the only operation that leaves every term of the 
 series unchanged is clearly the identical operation. 
 
 To each operation of the form 
 
 x = ax + f3 
 
 therefore will correspond a single substitution performed on the p^ 
 symbols just written, so that to the product of two operations will 
 correspond the product of the two homologous substitutions. The 
 group is therefore simply isomorphic with a substitution group of 
 degree p'". Moreover since the linear congruence 
 
 x= ax + ^ (mod. p) 
 
 has only a single solution when a is different from unity, and none 
 when a is unity, every substitution except identity must displace all 
 or all but one of the symbols. The substitution group is therefore 
 doubly transitive '. 
 
 Ex. 1. Apply the method just explained to the actual construc- 
 tion of a doubly transitive group of degree 8 and order 56. 
 
 Ex. 2. Shew that the equations 
 
 A^ = l, -S^*"-^ = 1 , AS-^AS = S-'^AS", 
 
 whei'e n is such that a primitive root of the congruence, 
 
 i^*"-' -1 = (mod. 2), 
 
 1 The author h^s shewn {Messenger of Mathematics, Vol. xxv (1896), pp. 
 147—153) that the type of group considered in the text is the only type of doubly 
 transitive group of degree p™ and order p"* (p™ - 1) when m = 3 ; and that, when 
 m = 2 and2)>3, the same is true. When ro=2 and 2' = 3, there is one other type. 
 
158 SPECIAL TYPES OF TRANSITIVE GROUPS [112 
 
 satisfies the congruence 
 
 i^^ + i +1 = (mod. 2), 
 
 suffice to define a group which can be expressed as a doubly transi- 
 tive group of degree 2" and order 2" (2"*- 1). 
 
 {Messenger of Mathematics, Vol. xxv, p. 189.) 
 
 113. A slight extension of the method of the preceding para- 
 graph will enable us to shew that, for every prime p, it is possible to 
 construct a triply transitive group of degree p^ + I and order 
 {p'^ + I) p"^ {p'^ - 1). The analysis of this group will form the 
 subject of investigation in Chapter XIV ; here we shall only demon- 
 strate its existence. 
 
 In the place of the operations of the last paragraph, we now 
 consider those of the form 
 
 where again a, /3, -y, 8 are powers of i, limited now by the condition 
 
 that ah — /8y is not congruent to zero (mod. p). When this relation 
 
 is satisfied, the set of operations again clearly form a group. More- 
 
 i" 
 over if we represent ^ by oo for all values of x, any operation of this 
 
 group, when carried out on the set of quantities 
 
 «=, 0, i, i\ , iP"-', 
 
 will change each of them into another of the set; while no operation 
 except identity will leave each symbol of the set unchanged. Hence 
 the group can be represented as a substitution group of degree 
 p^+\. 
 
 Now 
 
 X -i" V- — %"■ x — v'i'- — i"' 
 
 is an operation of the above form, which changes the three symbols 
 i", i*, i" into i"', i*', i°' respectively; and it is easy to modify this form 
 so that it holds when or oo occurs in the place of i", etc. Hence 
 the substitution group is triply transitive, since it contains an 
 operation transforming any three of the p™' +\ symbols into any 
 other three. 
 
 On the other hand, if the typical operation keeps the symbol x 
 unchanged, then 
 
 yx'+{h-a)x-p = (mod. p), 
 
 and this congruence cannot have more than two roots among the 
 set of p'^+\ symbols. Hence no substitution of the group, except 
 identity, keeps more than two symbols fixed. 
 
 Finally then, since the group is triply transitive and since it 
 contains no operation, except identity, that keeps more than two 
 symbols fixed, its order must (§ 108) be {p-" + I) p^ {p"" -\). 
 
114] INTRANSITIVE GROUPS 159 
 
 114. An intransitive substitution group, as defined in § 104, 
 is one which does not contain substitutions changing a^ into 
 
 each of the other symbols a^, as, , a„ operated on by the 
 
 group. Let us suppose that the substitutions of such a group 
 
 change a^ into aj, as, , aj, only. Then all the substitutions 
 
 of the group must interchange these k symbols among them- 
 selves ; for if the group contained a substitution changing a^ 
 into ajc+i, then the product of any substitution changing aj into 
 a^ by this latter substitution would change a^ into aje+i. Hence 
 the n symbols operated on by the group can be divided into a 
 number of sets, such that the substitutions of the group change 
 the symbols of any one set transitively among themselves, but 
 do not interchange the symbols of two distinct sets. It follows 
 immediately that the order of the group must be a common 
 multiple of the numbers of symbols in the different sets. 
 
 Suppose now that tti, a^, , aj is a set of symbols which 
 
 are interchanged transitively by all the substitutions of a group 
 G of degree n. If for a time we neglect the effect of the 
 substitutions of G on the remaining n — k symbols, the group 
 G will reduce to a transitive group H of degree k. The group 
 G is isomorphic with the group H; for if we take as the substi- 
 tutions of G, that correspond to a given substitution of H, those 
 
 which produce the same permutation of the symbols 01,^2, , a^ 
 
 as that produced by the substitution of H, then to the product 
 of any two substitutions of G will correspond the product of 
 the corresponding two substitutions of H. The isomorphism 
 thus shewn to exist may be simple or multiple. In the former 
 case, the order of H is the same as that of G; in the 
 latter case, the substitutions of G which correspond to the 
 identical substitution of H, i.e. those substitutions of G which 
 
 change none of the symbols o^, a^, , aj, form a self-conjugate 
 
 sub-group. 
 
 We will consider in particular an intransitive group G 
 which interchanges the symbols in two transitive sets; these 
 we will refer to as the a's and the /3's. Let Ga and G^ be the 
 two groups transitive in the a's and /3's respectively, to which 
 G reduces when we alternately leave out of account the effect 
 of the substitutions on the /S's and the a's. Also let g^ and g^ 
 
160 INTRANSITIVE [114 
 
 be the self-conjugate sub-groups of G, which keep respectively 
 all the /8's and all the a's unchanged; and denote the group 
 {g^, g^\ by g. This last group g, which is the direct product 
 of g^ and g^, is self-conjugate in Q, since it is generated by the 
 two self-conjugate groups g^ and g^. Now g^, is self-conjugate 
 not only in G but also in (?„; for G^ permutes the «'s in the 
 same way that G does, while any substitution of g^, not 
 affecting the yS's, is necessarily permutable with every sub- 
 stitution performed on the /3's. The group G^ is simply 
 
 C C 
 
 isomorphic with the group — , and G^ with — ; hence, using ng 
 
 to denote the order of a group H, 
 
 Let the substitutions of G be now divided into sets in respect 
 of the self-conjugate sub-group g, so that 
 
 G = g,Sg,Tg, 
 
 The group — is defined by the laws according to which 
 
 these sets combine among themselves, the sets being such that, 
 if any substitution of one set be multiplied by any substitution 
 of a second set, the resulting substitution will belong to a 
 definite third set. 
 
 If we now neglect the effect of the substitutions on the 
 symbols /3, the group G reduces to G^ and g reduces to g^, and 
 hence 
 
 Ga = ga, S^g^, Tag^., , 
 
 where S^, T„,, represent the substitutions 8, T, , so far 
 
 as they affect the a's. Moreover the substitutions in the different 
 sets into which G^ is thus divided must be all distinct since, 
 by the preceding relations between the orders of the groups, 
 
 their number is just equal to the order of Ga- Hence -^ is 
 
 defined by the laws according to which these sets of substi- 
 tutions combine. But if 
 
 Sg.Tg=Ug, 
 then necessarily , S^ga ■ T^a = U^a, 
 
116] GROUPS 161 
 
 and therefore, finally, the three groups —, -^, and — are simply 
 
 9 ffo. 9p 
 
 isomorphic. 
 
 C C 
 
 115. The relation of simple isomorphism between ~ and — 
 
 9a 9p 
 
 thus arrived at establishes between the groups Go, and G^ an 
 isomorphism of the most general kind (§ 32). 
 
 To every operation of Ga correspond rig operations of 
 Gfi, and to every operation of Gp correspond n^^ operations 
 of Go, ; so that to the product of any two operations of G^, (or G^) 
 there corresponds a definite set of Ug^ operations of G^ (or Ug^ 
 operations of G^). 
 
 Keturning now to the intransitive group G, its genesis from 
 the two transitive groups Ga and G^, with which it is isomorphic, 
 may be represented as follows. The Ug to Ug correspondence, 
 such as has just been described, having been established 
 between the groups Ga and Gp, each substitution of Ga is 
 multiplied by the rig substitutions that correspond to it in Gp. 
 The set of riff rig substitutions so obtained form a group, for 
 
 where, if 8^, 8^' are substitutions corresponding to 8a, 8a', then 
 8p" is a substitution corresponding to 8a"- Moreover, this 
 group may be equally well generated by multiplying every one 
 of the substitutions of Gp by the rig corresponding substitutions 
 of Ga ; and by a reference to the representations of G, Ga, and Gp, 
 as divided into sets of substitutions given above, it is imme- 
 diately obvious that all these substitutions occur in G. Hence, 
 as their number is equal to the order of G, the group thus 
 formed coincides with G. 
 
 116. The general result for any intransitive group, the 
 simplest case of which has been considered in the two last 
 paragraphs, may be stated in the following form : — 
 
 Theorem VI. 1/ G is an intransitive group of degree n 
 which permutes the n symbols in s transitive sets, and if (i) Gr is 
 what G becomes when the substitutions of G are performed on 
 
 B. 11 
 
162 INTilANMTIVE [116 
 
 the rth set of symbols only, (ii) F^ is what G becomes when the 
 
 substitutions of G are performed on all the sets except the rth, 
 
 (iii) gr is that sub-group of G which changes the symbols of the 
 
 rth set only, (iv) y^ is that sub-group of G which keeps all the 
 
 G V 
 
 symbols of the rth set unchanged : then the groups — and — 
 
 are simply isomorphic, and nr, Vr being the orders of g^, jr, <*« 
 Ur to Vr correspondence is thus established between the substi- 
 tutions of the groups Gr and V^. Moreover, the substitutions of 
 G are given, each once and once only, by multiplying each 
 substitution of Gr by the Vr substitutions of Vr that correspond to 
 it\ 
 
 It is not necessary to give an independent proof of this 
 theorem, since if, in the discussion of the two preceding para- 
 graphs, G^, Gp, go,, gt, g be replaced by Gr, Tr, gr, jr, {gr, 7r}. it 
 will be found each step of the process there carried out may 
 be repeated without alteration. 
 
 If we regard Ga and Gp as two given transitive groups 
 in distinct sets of symbols, the determination of all the 
 intransitive groups in the combined symbols, which reduce 
 to Ga or Gp when the symbols of the second or first set 
 are neglected, involves a knowledge of the composition of 
 the two groups. To each distinct m to n isomorphism, that 
 can be established between the two groups, there will corre- 
 spond a distinct intransitive group. If Ga is a simple group, 
 containing therefore only itself and identity self-conjugately, 
 then to each substitution of Gp there must correspond either 
 one or all of the substitutions of Ga ; and the former can be 
 the case only when Gp contains a self-conjugate sub-group H, 
 
 such that -^ is simply isomorphic with Ga- Hence, if the 
 
 order of Gp is less than twice the order of Ga, the only possible 
 intransitive group is the direct product of Ga and Gp, unless Qp 
 is simply isomorphic with Ga- 
 
 1 On intransitive groups, reference may be made to Bolza, " On the 
 construction of intransitive groups," Amer. Journal, Vol. xi, (1889), pp. 195 — 
 214. The general isomorphism which underlies the construction of these 
 groups is considered by Klein and Fricke, Vorlesungen iiber die Theorie der 
 elliptischen Modulfunctionen, Vol. i, (1890), pp. 402 — 406. 
 
117] ■ GROUPS 163 
 
 117. In illustration of T'-the p?eceding paragraphs, we will 
 determine the number of distinct- i^tifeuniitive groups of degree 7, 
 when the symbols are interchanged in transitive sets of 4 and 3 
 respectively. The four symbols will be referred to as the a's, and 
 the three symbols as the jS's. 
 
 If a transitive group of degree 4 contains operations of order 3, 
 it must be either the symmetric or the alternating group. If it 
 contains no operation of order 3, its order must be either 8 or 4. 
 By Sylow's theorem, the symmetric group of degree 4, being of 
 order 24, contains a single set of conjugate sub-groups of order 8, so 
 that there is only one type of transitive group of degree 4 and order 
 8. This is given by 
 
 ;S' = (1234), T={U), 
 
 so that S'=\, T'=l, TST=S". 
 
 This group contains three self-conjugate sub-groups of order 4, 
 namely, 
 
 (i) 1, (1234), (13) (24), (1432), 
 
 (ii) 1, (12) (34), (13) (24), (14) (23), 
 
 (iii) 1, (13) , (13) (24), (24) . 
 
 The two latter are simply isomorphic ; but as substitution groups, 
 they are of distinct form. Hence for ff„ in the construction of the 
 intransitive group, we may take either 6*341 the symmetric group of 
 the a's, or G^2t the alternating group of the a's, or G^, the above 
 group of order 8, or finally Gi, Gl, G^', the above three groups of 
 order 4. The only transitive groups of 3 symbols are G^, the sym- 
 metric group, and G^, the alternating group : one of these must be 
 taken for G^. 
 
 I. G, = G., 
 
 24- 
 
 The only self -conjugate sub-groups of G^ are G^is and G^. 
 
 Q 
 
 Also 7^ is a group of order 2 ; and it may easily be verified 
 6-12 
 
 that 7^ is simply isomorphic with the symmetric group of degree 3. 
 Gi 
 
 Hence (i) if ^,3 is G^, we may take 
 
 9a=G^, ffp = Gs, so that Mg=144, 
 
 9a = Gi2, 9p = G3, „ ng= 72, 
 
 and (ii) if G^ is G^, we must take 
 
 ga = Gu, 9^ = G„ giving ns = 72. 
 
 11—2 
 
164 INTRANSITIVE [117 
 
 II. G. = G,,. 
 
 Q 
 
 The only self- conjugate sub-group of G^jj is (?/; and 7^ is a 
 cyclical group of order 3. 
 
 Hence (i) if G^ is G^, we must take 
 
 go. = <?i2, 9^ = G^, giving We= 72 ; 
 and (ii) if Gphi Gg, we may take 
 
 9a = G^, 9^ = Gs, giving ?ig=36, 
 or S'a=G^4', S'p=l, ,, wg=12. 
 
 III. (?a=G«8- 
 
 The self-conjugate sub-groups of Gg, of order 4, are determined 
 above. 
 
 If (i) G^ is G^5, we may take 
 
 9a=Gs, 9p = Gs, giving ng=4:8, 
 
 9a = Gi, g^ = Gg, „ »iff=24, 
 
 9a = G/, g^ = Gs, „ Wg=24, 
 
 9a=Gi", 9^ = Gs, „ Jig = 24. 
 
 The two latter groups are simply isomorphic ; but regarded as 
 substitution groups, they are of distinct forms. 
 
 If (ii) Gp is Gg, we must take 
 
 S'a=<^8, 9^ = Gs, giving ?ig=24. 
 
 IV. G^=G,. 
 If (i) Gp is (xg, we may take 
 
 9a = Gi, g^ = Gs, giving n^ = 24, 
 or 9'a=(?2, 9p = Gg, „ ng=U; 
 
 G2 representing the group {1, (13) (24)}. 
 If (ii) Gp is Gg, we must take 
 
 9a=Gi, 9^ = Gs, giving ng=l2. 
 
 V. g,=g:. 
 
 There are, exactly as in the last case, three possibilities : <?/ taking 
 the place of G^. These groups are not however simply isomorphic 
 with the preceding three. 
 
 ». VI. Ga=G^". 
 
 There are in this case four possibilities. Of these three corre- 
 spond to those of IV, Gi" taking the place of G^. The remaining 
 one is given by G„ = Gg, g„ = G^, ^^ = G^, where G^ represents the 
 group {1, (13)}. Regarded as substitution groups all these are of 
 distinct form from the groups of V. 
 
118] GROUPS 165 
 
 There are thus SS' distinct intransitive substitution groups of 
 degree 7, in which the symbols are interchanged transitively in two 
 sets of i and 3 respectively. 
 
 118. Let v,.(r=0, I, 2, n) be the number of substi- 
 tutions of a group of degree n and order iV which leave exactly 
 r symbols unchanged, so that 
 
 r=n 
 r=0 
 
 Suppose first that the group is transitive; and in a sub- 
 group, which keeps one symbol unchanged, let v/ (r = 1, 
 
 2, , w) be the number of substitutions that leave exactly r 
 
 symbols unchanged, so that 
 
 N '■=» , 
 
 — = 2, Vr- 
 n r=l 
 
 Each of the n sub-groups, which leave a single symbol 
 unchanged, have v/ substitutions which leave exactly r symbols 
 unchanged; and each of these substitutions belong to r sub- 
 groups which leave one symbol unchanged. Hence 
 
 nvr = rvr, 
 
 r=n 
 
 and therefore N= S rvr', 
 
 or the number of unchanged symbols in all the substitutions 
 of a transitive group is equal to the order of the group. 
 
 Suppose, next, that the group is intransitive ; and consider a 
 
 set of s symbols among the n, which are permuted transitively 
 
 among themselves by the operations of the group. Let N^ be 
 
 the order of the self-conjugate sub-group Hi, which leaves 
 
 unchanged each of this set of s symbols. Then if we consider 
 
 the effect of the substitutions on this set of s symbols only, the 
 
 N . . 
 
 group reduces to a transitive group of order -^ with which the 
 
 original group is multiply isomorphic. If S' is any substitution 
 
 N 
 of this group of order ^ , and if SH^ denote the corresponding i\^i 
 
 -"1 
 operations of the original group, then every substitution of the 
 set SHi produces the same effect on the s symbols that 8' 
 
166 INTRANSITIVE [118 
 
 produces. Now the number of unchanged symbols in all the 
 substitutions of the transitive group of degree s and order -^ 
 
 . jsr 
 
 IS -=^ ; therefore, in all the substitutions of the original group, 
 
 the number of sjonbols of the set of s that remain unchanged is 
 N. The same reasoning applies to each separate set of the n 
 symbols, which are permuted transitively among themselves by 
 the operations of the group. Hence if there are ti such transi- 
 tive sets, the total number of symbols which remain unchanged 
 in all the substitutions of the group is Nti ; or 
 
 m, = 1, rvr, 
 
 119. The formula just obtained is the first of a series of 
 similar formulae, due to Herr Frobenius^ which are capable of 
 many useful applications. 
 
 Consider the symbol ((h., cti, , Oj,), where the letters are 
 
 p distinct letters chosen from the n which are operated on by 
 the group, the sequence in which the p letters are arranged 
 being regarded as essential. The number of such symbols 
 
 that can be formed from the n letters is w (m — 1) (n — p + 1). 
 
 Every substitution of the group interchanges this set of 
 symbols among themselves ; and no substitution can leave one 
 of the symbols unchanged unless it leaves each of the letters 
 forming it unchanged. Moreover no substitution of the group, 
 other than identity, can leave each of the set of symbols 
 unchanged. Hence the group can be expressed as a substitu- 
 tion group in this set of n(n—l) (n—p + l) symbols. 
 
 Every substitution of the group which leaves exactly r letters 
 unchanged will, if r <p, leave none of the set of symbols 
 unchanged ; while itr^p, it will leave exactly 
 
 r (r — 1) (r — p + 1) 
 
 unchanged. Hence, if the set of symbols are interchanged by 
 the substitutions of the group in tp transitive sets, then 
 
 mp=-z 
 
 r=p '' P- 
 
 1 " Ueber die Congruenz naoh eiuem aus zwei eudlichen Gruppen gebildeten 
 Doppelmodul," Crelle, t. ci, (1887), p. 288. 
 
119] GROUPS 167 
 
 From the symbol (ai, a^, , ap), containing p letters, 
 
 may be formed n — p symbols containing p + 1 letters, by 
 adding any one of the remaining n — p letters in the last place. 
 
 If the symbol (aj, a,,, , a^) is one of a transitive set of 
 
 s symbols, to these there will correspond s(n — p) symbols of 
 p + 1 letters. No symbol oi p + 1 letters, which is not included 
 among these s(n —p) symbols, can enter into a transitive set 
 with any one of the s(n — p); since if it did, the s symbols oip 
 letters would not form a transitive set. Hence the s(n — p) 
 symbols must form, by themselves, a number of transitive sets 
 of the symbols of p + 1 letters ; and this number clearly cannot 
 be less than 1 nor greater than n — p. Accordingly, to every 
 transitive set of the symbols of p letters, there correspond 
 x(l^x^n—p) transitive sets of the symbols of ^ + 1 letters; 
 and therefore 
 
 tp^tp+i:^{n-p)tp. 
 
 Ex. 1. li tp and «j,+i are equal and if ^ <w - 1, shew that ^^^j is 
 unity and that the group is (p + l)-ply transitive. 
 
 Ex. 2. Apply the method of § 119 to shew that no substitution, 
 except identity, of a ^-ply transitive group, which does not contain 
 the alternating group, can displace less than 2A — 2 letters. 
 
 Note to § 108. 
 
 The results of M. Jordan, stated on p. 149, may be established 
 as follows. Let G he a, quadruply transitive group of degree n and 
 order n(n-l){n-2)(n- 3), so that m - 2 is the power of a prime. 
 There is a single operation of G which changes any four symbols 
 into any other four symbols. Let a, b, c, d be four of the symbols 
 operated on by G. The operations of G which permute these 
 symbols among themselves form a sub-group H, simply isomorphic 
 with the symmetric group of degree 4. Suppose that the remaining 
 w-4 symbols are permuted by If in transitive sets of n^, n^, ... 
 symbols each. 
 
 The only groups with which IT is multiply isomorphic are 
 (i) the symmetric group of degree 3, (ii) a group of order 2. If 
 then, when we consider the effect of -ff on a set of % symbols which 
 are permuted transitively by it, the group of degree ji^ so obtained 
 is one with which H is multiply isomorphic, this group must be 
 
168 NOTE TO § 108 
 
 either the symmetric group of degree 3 or a group of order 2. 
 Hence n^ must be either 6, 3 or 2. Since in this case H will 
 contain operations leaving all the n^ symbols unchanged, the value 
 6 for Wi is inadmissible. If n^ is 3, the operations of S, which 
 give the substitutions 
 
 1, (ab) (cd), (ad) (be), {ac) (bd), 
 
 of a, b, c, d, leave the 3 symbols unchanged ; and if n^ is 2, the 
 operations of H which give all the even substitutions of a, b, c, d, 
 leave the 2 symbols unchanged. 
 
 If, on the other hand, when we consider the effect of H on the 
 set of Wj symbols only, the transitive group of degree n^ so obtained 
 is simply isomorphic with S, then % must be either I, 6, 8, 12 or 
 24. In this case, the value 4 for Wj is inadmissible. In fact, if 
 rii were 4 the operation of S which leaves c and d unchanged 
 would also leave two of the ti^ symbols unchanged; and this is 
 impossible since no operation of H, except identity, can leave more 
 than 3 symbols unchanged. Por a similar reason, the value 12 for 
 1% is inadmissible ; whUe, if % is 6, the sub-group that keeps one of 
 the Wi symbols unchanged must be cyclical. 
 
 The only other possible value for ti^ is unity. 
 
 Suppose, first, that mj is 2. If any of the remaining numbers 
 n^, n^, ... differ from 24, it may be shewn immediately that 
 S contains operations which keep more than 3 symbols fixed. 
 Hence in this case n is congruent to 6 (mod. 24) ; and since ji - 2 
 cannot then be a power of a prime unless n = Q, this case gives the 
 alternating group of degree 6. 
 
 Next, suppose that % is 3. The only admissible values for 
 n^, rig, ... are 8 and 24. In this case, the sub-group of E which 
 keeps all the v^ (= 3) symbols unchanged is a non-cyclical sub-group 
 of order 4. But we have seen in § 105 that, if ra-3 is even, a 
 sub-group of order n-3 which keeps 3 symbols unchanged can only 
 have a single operation of order 2. Hence this case cannot occur. 
 
 Next, suppose that n^ is 8. The only admissible values for 
 Wj, Wg, ... are again 24. This case cannot occur since no number 
 congruent to 10, mod. 24, can be a power of a prime. 
 
 The only remaining possibilities are : 
 
 (i) 
 
 Wl=l, 
 
 _ 
 
 . =24, 
 
 (ii) 
 
 »»2 = W3= 
 
 . =24, 
 
 (iii) 
 
 n, = 6, 
 
 Wa = Wg= 
 
 . =24, 
 
 (iv) 
 
 71^=1, 
 
 ^2 = 6, Ms = . 
 
 .. = 24, 
 
 (V) 
 
 %=1, 
 
 Wa = 6, Ws = , 
 
 ..=0, 
 
 (vi) 
 
 Wl=l, 
 
 «2 = «3 = • 
 
 ..=0. 
 
 The first of them cannot occur, since then ti - 2 is not the power 
 
NOTE TO § 108 169 
 
 of a prime. In the second case, an operation of H which leaves c 
 and d unchanged would be of the form 
 
 («6)(a^)(y8) , 
 
 where there are |^(w-3) independent transpositions; while an 
 operation of H, of order 2, which leaves none of the symbols a, h, 
 c, (f unchanged, will consist of the product of ^{n-l) independent 
 transpositions. Now the operation 
 
 (ab) {aP){y8) 
 
 occurs in the group, conjugate to H, which permutes a, b, a, fi, 
 among themselves ; and as an operation of this group, it must be 
 the product of |(n— 1) independent transpositions. This is a 
 contradiction : hence this case cannot occur. 
 
 In case (iii), let 1, 2, 3, 4, 5, 6 be the six symbols that are 
 permuted transitively. The self-conjugate sub-group of H of order 
 
 4 will consist of identity and the three substitutions 
 
 (a6)(cc?)(34)(56)(a;8)(y8) , 
 
 (ac)(6^)(56)(12)(ay)(/38) , 
 
 M(&c)(12)(34)(a8)(^y) 
 
 This sub-group will also occur as the self-conjugate sub-group of 
 order 4 of the group, conjugate to H, which permutes a, ^8, y, S 
 among themselves. If S, S' are two operations, of order 3, of these 
 sub-groups which give the same permutation of 1, 2, 3, 4, 5, 6, then 
 SS''''- is an operation of G, distinct from identity, which keeps the 
 six sjTnbols unchanged. Hence this case cannot occur. Precisely 
 the same reasoning applies to case (iv). 
 
 Finally then, cases (v) and (vi), in which n is respectively 1 1 and 
 
 5 are the only remaining possibilities. 
 
 When nis 5, G is the symmetric group of degree 5. 
 
 When n is 11, the group G, if it exists, is of degree 11 and 
 order 11.10.9.8. A sub-group of order 8 which keeps three 
 symbols fixed must contain a single operation of order 2 ; hence 
 it must either be cyclical or of the type given in Theorem V, § 63. 
 When expressed in 8 symbols, this is generated by 
 
 (1254) (3867) and (1758) (2643), 
 
 while we may take a cyclical group of order 8 to be generated by 
 
 (12835476). 
 
 It is easy to verify that each of these substitutions transforms the 
 group of order 9, generated by 
 
 (123) (456) (789) and (147) (258) (369), 
 into itself. 
 
 If we now apply the method of § 109, we find that each of 
 these doubly transitive groups of degree 9 is contained in a triply 
 
170 NOTE TO § 110 
 
 transitive group of degree 10; a repetition of the same process of 
 trial shews that, while the group 
 
 {(123) (456) (789), (12835476)} 
 is not contained in a quadruply transitive group of degree 11 and 
 order 11 . 10 . 9 , 8, the group 
 
 {(123) (456) (789), (1254) (3867), (1758) (2643)} 
 and the two substitutions 
 
 (a2) (58) (64) (79), (al) (57) (68) (49) 
 actually generate such a group. 
 
 A further trial will shew that this group and the substitution 
 (6c) (47) (58) (69) 
 generate a group of degree 12 and order 12. 11. 10. 9. 8; but that 
 there is no group of degree 13 which contains the last group as the 
 sub-group that keeps one symbol fixed. 
 
 The three substitutions, of order two, just given generate that 
 sub-group of order 24, of the transitive group of degree 11 and 
 order 11 . 10 . 9 . 8 in the symbols 
 
 a, h, c, 2, 3, 4, 5, 6, 7, 8, 9, 
 which permutes a, b, c, 2 among themselves. 
 
 If A<t;4, a A-ply transitive group, of degree n and order 
 
 n{n— 1) {n — k+ 1), must contain a quadruply transitive group of 
 
 degree n-k + A and order {n — k + i:){n — h+i){n-k + 2){n-h + \). 
 Hence, if the group is neither the alternating group nor the sym- 
 metric group of degree n, it must be either the group of degree 11 
 or the group of degree 12 that have been determined above. 
 
 Note to § 110. 
 
 It may be shewn that, with a single exception when w = 12, the 
 inequality (p. 152) 
 
 kl^^n+l 
 may be replaced by 
 
 k<^n+ 1. 
 
 Since k is an integer, it is only necessary to consider the case in 
 which m is a multiple of 3, so that we may write 3m for n. If, in 
 this case, k = 7ii + \, then 2^ - 2 = 2m ; and a (m -I- l)-ply transitive 
 group of degree 3m, which does not contain the alternating group, 
 can therefore have no substitution which displaces fewer than 2m 
 
 symbols. Its order must therefore be 3m (3m — 1) (2m + 1) im. 
 
 From M. Jordan's results, which have just been proved, such a 
 group exists only when n= 12 ; and therefore when n is not 12, a 
 A-ply transitive group of degree n, which does not contain the 
 alternating group, can only exist if k^<n. + l. 
 
CHAPTEE IX. 
 
 ON SUBSTITUTION GEOUPS : PRIMITIVE AND 
 IMPRIMITIVE GROUPS. 
 
 120. We have seen that the symbols permuted by the 
 operations of an intransitive substitution group may be divided 
 ioto sets, such that every substitution of the group permutes 
 the symbols of each set among themselves. For a transitive 
 group the symbols must, from this point of view, be regarded 
 as forming a single set. It may however in particular cases 
 be possible to divide the symbols permuted by a transitive 
 group into sets in such a way, that every substitution of the 
 group either interchanges the symbols of any set among them- 
 selves or else changes them all into the symbols of some other 
 set. That this may be possible, it is clearly necessary that each 
 set shall contain the same number of symbols. 
 
 In the present Chapter we shall discuss those properties 
 of transitive groups which depend on their possessing or not 
 possessing the property indicated. 
 
 121. Definition. When the symbols operated on by a 
 transitive substitution group can be divided into sets, each set 
 containing the same number of distinct symbols and no symbol 
 occurring in two different sets, and when the sets are such 
 that all the symbols of any set are either interchanged among 
 themselves or changed into the symbols of another set by every 
 substitution of the group, the group is called imprimitive. 
 When no such division into sets is possible, the group is called 
 
172 IMPRIMITIVE SYSTEMS [121 
 
 primitive. The sets of symbols which are interchanged by 
 an imprimitive group are called imprimitive systems. 
 
 A simple example of an imprimitive group is given by 
 group VII of I 17, An examination of the substitutions of this 
 group will shew that they all either transform the systems of 
 symbols xyz and abc into themselves or else interchange them, 
 and that the same is true of the systems xa, yb, zc ; so that, in 
 this case, the symbols may be divided into two distinct sets of 
 imprimitive systems. 
 
 It follows at once, from the definition, that an imprimitive 
 group cannot be more than simply transitive. For if it were 
 doubly transitive, it would contain substitutions changing any 
 two symbols into any other two, and of these the first pair 
 might be chosen from the same imprimitive system and the 
 second pair from distinct systems. 
 
 The question as to whether a given group can be expressed 
 as a transitive group of given degree, and the further question 
 as to whether such a representation of the group, when possible, 
 is imprimitive or primitive, finds its complete solution in the 
 following investigation due to Herr Dyck^. 
 
 122. In § 20 it was shewn how any group of order N 
 could be represented as a substitution group of N symbols. 
 This form of the group, defined as the regular form, is simply 
 transitive ; for all its substitutions except identity displace all 
 the symbols, and therefore there must be just one substitution 
 changing a given symbol into any other. Let us now suppose 
 that N=iMv, and that has a sub-group H of order n, consisting 
 of the operations 
 
 Si{=\), S3, S3, , (S^, 
 
 so that every operation of the group can be represented uniquely 
 in the form 
 
 Sr^Tn (m = l, 2,3, ,/L6; 7i = l,2,3, ,v); 2\ = 1. 
 
 The tableau representing the group as a substitution group 
 of N symbols will, in terms of these symbols, take the form 
 
 1 Dyok, " Gruppentheoretisohe Studien, 11," Math. Ann. xxii, (1883), pp. 
 86—95. 
 
122] OF A REGULAR GROUP 173 
 
 given on the following page. Every symbol in this tableau is 
 of the form 
 
 Om-'-n^m'-'-n' ] 
 
 and such a symbol will belong to the column headed by SmTn 
 and to the line beginning with Sm'Tn'. The symbols in any 
 line (or column) differ in arrangement only from those in the 
 leading line (or column); hence Tic must occur in the line 
 beginning with Sm'Tn'- We may therefore suppose that 
 
 and then S^Sm TnSm'Tn' = S^Tji, 
 
 Since 1, 8^, Sg, ,8^ 
 
 form a group, these symbols differ from 
 
 Omj ^2^nii ^s^tnt ^fi^m 
 
 only in the sequence in which they are written ; and therefore 
 the set of symbols 
 
 8pT^8rn'T^. (p = l,2, , /.) 
 
 is identical, except as regards arrangement, with the set 
 
 8pT, (p = l, 2, ^). 
 
 Hence the substitution of G, represented by the line be- 
 ginning with 8m;Tn', changes the set of symbols 8pTn into the 
 set 8pT]e in some sequence or other. 
 
 Every substitution of the group therefore changes the 
 symbols of each of the v sets, into which the first line of the 
 tableau is divided, either into themselves or into the symbols of 
 some other of the v sets. Hence : — 
 
 Theorem I. If a group of order fxv contains a sub-group 
 of order fi, the regular form of the group will be imprimitive in 
 suck a way that the /mv symbols may be divided into v imprimi- 
 tive systems of (i symbols each. 
 
 The converse of this theorem is also true. For if the iiv 
 symbols 
 
174 
 
 IMPRIMITIVB SYSTEMS 
 
 [122 
 
 
 
 
 
 E.^^t 
 
 E? 
 
 E? 
 55* 
 Es' 
 
 =5" 
 
 E? 
 
 as" 
 
 : E? 
 
 
 
 =5 =5" 
 
 E.' 
 E.^ 
 
 i 
 
 E.' 
 
 S* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 is 
 
 
 
 !^cEh' 
 
 :e? 
 
 ^! 
 
 E? 
 
 Eh^ 
 
 E? 
 E? 
 
 
 
 ^^Eh'^ 
 
 ^cE."' 
 Eh5,« 
 
 E.^ 
 as* 
 Eh= 
 
 $ 
 
 a? 
 E-^ 
 
 i 
 
 aa" 
 
 E,^ 
 as* 
 E? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 as 
 
 r— 1 
 
 =5 
 
 1 
 
 ; 
 
 =1 
 
 3 
 
 
 
 6^ 
 
 e:^ 
 25 
 
 E^ 
 ss 
 
 E? 
 
 = E^ 
 =5 
 
 25 
 
 '?' 
 =5 
 
 E.' 
 
 5 
 
 E^ 
 
 =? 
 S5 
 
 E^ 
 
 i 
 
 =5 
 
 E.' 
 
 =5 
 
 e:; 
 
 at 
 
 3 
 4 
 
 
 
 E^ 
 as 
 
 E^ 
 =5 
 
 Eh' 
 
 E^ 
 
 lac 
 
 =5 
 
 e; 
 =5 
 
 E^ 
 '^ 
 
 =5 
 
 E^ 
 
 '3 
 
 E- 
 E- 
 
 Or 
 
 ar 
 E- 
 &: 
 
 E- 
 
 6C 
 
 i. 
 
 i. 
 
123] OF A EEGULAE GROUP 175 
 
 by whose permutations the group can be expressed in regular 
 form, are divisible into v imprimitive systems of /* each, let 
 
 be that system which contains identity. Then this system 
 and the systems 
 
 Sm, S^Sm, SaSm, , ^ij^Sm, Vl = 2, 3, fl, 
 
 having the symbol 8m in common, must have all their symbols 
 in common ; therefore the product of any two of the operations 
 of this set of fj. operations is another operation of the set. The 
 set therefore forms a group. 
 
 123. Let us now represent the imprimitive system 
 
 J-mi '^2-'m; "s-^m; > ^n-t-m 
 
 by the single symbol Um, for all values of m. If we then, in 
 the preceding tableau representing the group, pay attention only 
 to the way in which the systems are permuted among themselves, 
 without regarding the permutations of the symbols within the in- 
 dividual systems, we obtain a substitution group of the v symbols 
 
 u„u„ u„ , u,. 
 
 This group is isomorphic with the original group ; and if 
 no substitution of the original group interchanges among 
 themselves the symbols of each imprimitive system, the iso- 
 morphism must be simple. Now a substitution of Q, which 
 does not change any imprimitive system into another, must, if 
 it exists, be a substitution 8' of the sub-group H, which is 
 constituted by 
 
 such that Tm8' belongs to the set Um for each suffix m ; and 
 therefore, for each suffix m, we must have an equation 
 
 where 8" is another substitution of H. The sub-group H 
 must therefore contain every substitution of which is conju- 
 gate to 8': in other words, it must contain a self-conjugate 
 sub-group of 0. Hence : — 
 
 Theorem II. If H, of order fi, is a sub-group of Q of 
 order /iv, and if no self-conjugate sub-group of G is contained in 
 H, then G can be expressed as a transitive group of degree v. 
 
176 MODES OF REPRESENTING [123 
 
 That the converse of this theorem is true is immediately 
 obvious. 
 
 124. If the sub-group Zf of G is contained in a greater 
 sub-group K of order jmv, where v = v'v", the operations of K 
 consist of the sets 
 
 H, HTi, HTs, , HT^, 
 
 and the operations of G of the sets 
 
 K,KB„KB„ , ZiJ„"; 
 
 while the set KRm is made up of the sets 
 
 SRmt Sf^Rm! ST^Rm, , HT^'R^. 
 
 The regular form of the group has v'v" imprimitive systems 
 corresponding to the sub-group H, and v" imprimitive systems 
 corresponding to the sub-group K; the above method of re- 
 presenting K shews that each of the latter systems contain v 
 complete systems of the former set. 
 
 Now it has just been proved that, if H contains no self- 
 conjugate sub-group of 0, the group can be represented as a 
 transitive substitution group of the symbols 
 
 HTnRm {n=l,% ,v\ m = l, 2, ,v"). 
 
 But from the division of the symbols in the regular form of 
 the group into v" imprimitive systems, it follows that the set of 
 symbols 
 
 HT^R,^ (n = l, 2, ,/) 
 
 must either be permuted among themselves or be changed into 
 another set 
 
 HTr^R^. (n = l, 2, ,v'), 
 
 by every substitution of the group. The representation of the 
 group as a transitive group of v'v" symbols is therefore im- 
 primitive. Hence : — 
 
 Theorem III. If a group of order fiv'v" has a sub-group 
 H of order ix, which contains no self -conjugate sub-group of G; 
 and if H is contained in a sub-group K of of order iw; 
 then the representation of Q as a transitive group of degree 
 v'v" {in respect of H) is imprimitive, the v'v" symbols being 
 divisible into v" systems of v symbols each. 
 
125] A GROUP AS TRANSITIVE 177 
 
 Corollary I. A transitive group of order fiv and degree v 
 will be primitive if, and only if, a sub-group of order fi that 
 keeps one symbol fixed is a maximum sub-group. 
 
 Corollary II. A group, which contains other self-conjugate 
 operations besides identity, cannot be represented in primitive 
 form. 
 
 For if a sub-group IT of order fi contains a self-conjugate 
 operation, the group (of order fiv) cannot be represented as 
 a transitive group of degree v in respect of H; and if H 
 contains none of the self-conjugate operations, and is not a 
 self-conjugate sub-group, it cannot be a maximum sub-group. 
 
 In particular, a group whose order is the power of a prime 
 cannot be represented as a primitive group. 
 
 Corollary III. An Abelian group when represented as a 
 transitive substitution group, must be in regular form. 
 
 Corollary IV. A simple group can always be represented 
 in primitive form. 
 
 125. Every possible representation of a group as a 
 transitive substitution group is given by the method of the 
 preceding paragraphs. There is another method of dealing 
 with the same problem which we may shortly consider here in 
 view of its utility in many special cases, though it does not in 
 general lead to all possible modes of representation. Let 
 
 Hi, -02, , H^ 
 
 be a conjugate set of sub-groups (or operations) of a given 
 group G, and let 
 
 be a set of sub-groups of G, such that Ir is the greatest sub- 
 group containing H^ self-conjugately. The latter set of groups 
 are not necessarily all distinct ; in fact, we have seen in § 55 
 that, when the order of G is the power of a prime, they cannot 
 be all distinct. 
 
 If S is any operation of G, then 
 
 ( Si , H, , , H, \ 
 
 \S-'HiS, S-'H,S, , S-'H,S) 
 
 B- 12 
 
178 MODES OF REPRESENTING [125 
 
 is a substitution performed on the set of symbols Hi,!!^, ...,H • 
 and if for S each substitution of the group is written in turn 
 these substitutions form a transitive substitution group of 
 degree v. The substitution corresponding to S, followed by 
 the substitution 
 
 ( 
 
 
 corresponding to T, gives the substitution 
 
 -Si . -Ha , , H. 
 
 )■■ 
 
 and therefore the substitutions form a group isomorphic with 
 G, since the product of the substitutions corresponding to S and 
 T is the substitution corresponding to ST. 
 
 Moreover, since there are operations of which transform 
 Hj, into each of the other sub-groups (or operations) of the 
 conjugate set, the substitution group is transitive in the v 
 symbols. The substitution group will be simply isomorphic 
 with G if, and only if, there is no operation of G which 
 transforms each of the v sub-groups into itself Now the only 
 operations of G which transform Hi into itself are the oper- 
 ations of /i ; and hence the substitution group will be simply 
 isomorphic with G if, and only if, the conjugate set of sub- 
 groups 
 
 -'ij -'2) > -'l') 
 
 has no common sub-group except identity. This will be the 
 case only when /i contains no self-conjugate sub-group of 0. 
 
 It has been seen (§ 123) that, when this condition is satisfied, 
 G can be represented as a transitive substitution group whose 
 degree is v, the ratio of the orders of G and Ii. That the 
 form there obtained is identical with the form obtained in the 
 present paragraph may be easily verified. Thus in the earlier 
 form, the substitution corresponding to 8 is 
 
 ■'l-'l) -'l-'2) ) -'l-'l'A /7T_-|\ 
 
 A ZS, I,T,8, , A T,S) ' ^^'-'■>' 
 
126] A GROUP AS TRANSITIVE 179 
 
 or in abbreviated form 
 
 
 if T,S=iT,; 
 
 i being some operation of 7i . 
 
 In the present mode of representation, the substitution 
 
 corresponding to S can be written in the form 
 
 / TrH.T, , Tr'H.T, , T-'H,T^ \ 
 
 \8-'Tc'H,T,8, S-'Tr'H,TA , S-'Tr'H,T,S) ' 
 
 since each operation of the set /iT„ will transform H^ into the 
 
 same sub-group. Now, if a corresponding abbreviated form be 
 
 used, this substitution may be written 
 
 and therefore the symbols J, T^ in the one form are permuted 
 by the substitutions in identically the same manner as the 
 corresponding symbols T^'^H^Tx in the other form. 
 
 It should be noticed that, if contains self-conjugate opera- 
 tions other than identity, these operations necessarily occur 
 in /i ; and therefore in such a case the present method cannot 
 lead to a substitution group which is simply isomorphic with 
 0. In any case, if K is the greatest self-conjugate sub-group of 
 contained in /,, the substitution group is simply isomorphic 
 
 with ^. 
 
 126. As an illustration of the preceding paragraphs, we will 
 determine the different modes in which the alternating group of 
 degree 5 can be represented as a transitive group. 
 
 The only cyclical sub-groups contained in G^f,, the alternating 
 group of degree 5, are groups of orders 2, 3 and 5 ; and of each of 
 these cyclical- sub-groups there is a single conjugate set. 
 
 The non-cyclical sub-groups may be determined as follows. The 
 lowest possible order for such a sub-group is 4 ; since this is the 
 highest power of 2 that divides 60, there is a single conjugate set of 
 sub-groups of order 4. The next lowest possible order is 6. Now 
 no operation of order 3 is permutable with an operation of order 
 2, as the group contains no operations of order 6 ; on the other hand, 
 every sub-group of order 3 is permutable with an operation of order 
 2; thus 
 
 (12) (45) (123) (12) (45) = (132). 
 
 12—2 
 
180 MODES OF REPRESENTING A GROUP AS TRANSITIVE [126 
 
 There is therefore a single set of conjugate sub-groups of order 6. 
 The next lowest possible order is 10. The group contains no opera- 
 tion of order 10 ; but everj^ sub-group of order 5 is permutable with 
 an operation of order 2 ; thus 
 
 (14) (23) (12345) (14) (23) = (15432). 
 
 There is therefoi-e a single conjugate set of sub-.groups of order 
 10. The next lowest possible order is 12. If the group contains a 
 sub-group of this order, it must be transitive in 4 symbols. N6w the 
 alternating group of 4 symbols is of order 12. Hence (?jj must 
 contain a single conjugate set of sub-groups of order 12. The only 
 other possible orders are 15, 20 and 30. The reader will readily 
 verify directly that there are no sub-groups of these orders. This 
 can also be seen indirectly, since G^^ is a simple group and therefore, 
 if there were a sub-group of order 15, the group could be expressed 
 transitively in 4 symbols. Since the group contains operations of 
 order 5, this is clearly impossible. 
 
 Hence finally, since each of the sub-groups leads to a transitive 
 representation of the group, G^^ can be represented as a transitive 
 substitution group in 30, 20, 15, 12, 10, 6 and 5 symbols, and in 
 one distinct form in each case. The second method, as given in 
 § 125, does not lead to all these modes of representation. The 
 group will be found to contain 15 conjugate operations (or sub- 
 groups) of order 2 : lO'^onjugate sub-groups and 20 conjugate 
 operations of orcisr 3 : 5 conjugate sub-groups of order 4 : 6 con- 
 jugate sub-groups and two sets of 12 conjugate operations, each of 
 order 5 : 10 conjugate sub-groups of order 6 : 6 conjugate sub-groups 
 of order 10 : and 5 conjugate sub-groups of order 12. Hence, by 
 using the second method, the representation of the group as transi- 
 tive in 30 symbols would be missed. 
 
 Since a sub-group of order 2 is contained in sub-groups of orders 
 4, 6, 10 and 12, the 30 symbols permuted by ffjj, when it is ex- 
 pressed as a transitive group of degree 30, can be divided into sets 
 of imprimitive systems, containing respectively 2, 3, 5 and 6 symbols 
 each. Similarly, when G^^ is represented as a transitive group of 
 degree 20, 15 or 12, it is imprimitive. When expressed as a group 
 of order 10, 6 or 5, it is primitive. 
 
 127. As a further illustration, we shall determine all the 
 distinct forms of imprimitive groups of degree 6. Let G be such 
 a group, and H that sub-group of G which interchanges among 
 themselves the symbols of each imprimitive system. 
 
 We will first suppose that there are two systems of three symbols 
 each, viz. 
 
 1, 2, 3 and 4, 5, 6. 
 
127] IMPRIMITIVE GROUPS OF DEGREE SIX 181 
 
 In this case, H cannot consist of the identical operation only ; 
 for there must be a substitution changing 1 into 2, and this must 
 permute 1, 2 and 3 among themselves, and therefore also 4, 5 and 6 
 among themselves. 
 
 Let H contain substitutions which leave 4, 5 and 6 unchanged. 
 These must (§ 114) form a self-conjugate sub-group of H, which will 
 be either 
 
 {(123), (12)} or {(123)}. 
 
 Since G has substitutions interchanging the systems, E must 
 similarly contain 
 
 {(456), (45)} or {(456)}. 
 
 In the first alternative, H is the group 
 
 (i) {(123), (12), (456), (45)}; 
 
 for H contains this gi"oup, and on the other hand, this is the most 
 general group that interchanges the six symbols in two intransitive 
 systems of three each. The order of this group is 2^ . 31 
 
 In the second alternative, ZT contains the self-conjugate sub-group 
 
 {(123), (456)}. 
 
 Now if H is of order 2^.3^, it is necessarily of the form (i). If 
 it is of order 2 . 3", it must contain a substitution of order 2 which 
 transforms the sub-group just given into itself. This may be taken, 
 without loss of generalitv, to be (12) (45) ; and then H is the 
 group • '•' 
 
 '(ii) {(123), (456), (12) (45)}. 
 
 If H is of order 3^, it is the group 
 
 (iii) {(123), (456)}. 
 
 Next, let H contain no substitutions which leave 4, 5 and 6 
 unchanged. Then H is simply isomorphic with a group of degree 
 3, and therefore it must be of the form 
 
 (iv) {(123) (456), (12) (45)}, 
 
 or (v) {(123) (456)}. 
 
 Now ^ is of order 2 : therefore G must have a substitution of 
 
 order 2 or 4, which interchanges the systems. If the order of H is 
 odd, this substitution must be of order 2. When the substitutior 
 is of order 2, we may, without loss of generality, take it to be 
 (14) (25) (36) or (14) (26) (35). If H is of the form (i), (ii), or (iv), 
 we get for G, in each case, the same group whichever of these substi- 
 tutions we take. When H is of the form (iii), we get two groups which 
 are easily seen to be conjugate in the symmetric group. These we 
 do not regard as distinct. When H is of the form (v), we get two 
 
182 IMPEIMITIVE GROUPS OF DEGREE SIX [127 
 
 distinct groups, one of which is simply isomorphic with the sjm.- 
 metric group of three symbols, while the other is a cyclical group. 
 In these two latter cases, the symbols can be divided into three im- 
 primitive systems of two each. 
 
 If the substitution which interchanges the systems is of order 4, 
 its square must occur in H ; we may therefore take it to be 
 (1426) (36). When H is of form (i), this gives the same group as 
 before ; but when S is of either of the forms (ii) or (iv), we get new 
 forms for G. There are therefore ©igbt distinct forms of groups of 
 degree 6, in which the symbols form two imprimitive systems of 
 three symbols each. 
 
 Secondly, suppose that there are three systems of symbols, 
 containing two each, viz. 
 
 1, 2; 3, 4; and 5, 6. 
 
 The self -conjugate sub-group H is of order 2', 2% 2 or 1. 
 Corresponding to the first three cases, the forms of H are easily 
 seen to be 
 
 (i) {(12), (34), (56)}, 
 
 (ii) {(12)(34), (34)(56),(12)(56)}, 
 
 and (iii) {(12) (34) (56)}. 
 
 Q 
 
 Again, since -^ interchanges the three systems, it must be 
 
 simply isomorphic with a group of degree 3, and its order is therefore 
 either 3 or 6. First, let its order be 3. It must then contain a 
 substitution of order 3 which, without loss of generality, may be 
 taken to be (135) (246) ; this gives, with the three above forms of H, 
 three distinct forms for G. The form of G corresponding to the form 
 (iii) of H is, however, the same as one of those already determined. 
 
 Q 
 
 If -jf is of order 6, G must contain as a self-conjugate sub-group 
 
 ■ £1 
 
 Q 
 
 one of the three groups just obtained. Also if -jj were cyclical, 
 
 there would be a substitution in G, not belonging to H and 
 permutable with (135) (246). This is clearly impossible, and there- 
 fore G must contain a substitution which transforms (135) (246) 
 into its inverse. We may take this to be (13) (24) or (14) (23) (56). 
 With the form (i) for H, these two substitutions lead to the same 
 group. When H is of the form (ii), they give two distinct forms 
 for G. When H is of the form (iii), G admits the imprimitive 
 systems 1, 3, 5, and 2, 4, 6. 
 
 Lastly, if H is the identical operation, G is necessarily of order 6 ; 
 no new forms can arise. 
 
128] TEST OF PRIMITIVITY 183 
 
 There are therefore five distinct forms of groups of degree 6, in 
 which the symbols form three imprimitive systems of two symbols 
 each but do not at the same time form two imprimitive systems of 
 three symbols each. 
 
 128. An actual test to determine whether any transitive 
 group is primitive or imprimitive may be applied as follows. 
 Consider the effect of the substitutions of the group G* on r of 
 the symbols which are permuted transitively by it. Those 
 substitutions, which permute the r symbols, say 
 
 Oj\> (^2 3 ^r 
 
 among themselves, form a sub-group H. Now suppose that 
 every substitution, which changes a^ into one of the r symbols, 
 belongs to H. Then if /S is a substitution, which does not 
 permute the r symbols among themselves, it must change 
 them into a new set 
 
 bi, h, , K, 
 
 which has no symbol in common with the previous set; and 
 every operation of the set HS changes all the as into b's. 
 Moreover, since G is transitive, S must permute the a's 
 transitively ; and therefore the set HS must contain substitu- 
 tions changing ai into each one of the b's. 
 
 Suppose now, if possible, that the group contains a substi- 
 tution S', which changes some of the a's into b's, and the 
 remainder into new symbols. We may assume that S' changes 
 Oj into 6], and ag into a new symbol Cg. Among the set HS 
 there is at least one substitution, T, which changes aj into b^. 
 Hence TS'~^ changes Oj into itself and some new symbol into 
 a^. This however contradicts the supposition that every 
 substitution, which changes a^ into one of the set of a's, 
 belongs to H. Hence no substitution such as S' can belong to 
 Q; and every substitution, which changes one of the a's into 
 one of the b's, must change all the a's into b's. 
 
 If the substitutions of the group are not thus exhausted, 
 there must be another set of r symbols 
 
 which are all distinct from the previous sets, such that some 
 substitution changes all the a's into c's. We may now repeat 
 
184 TEST OF PRIMITIVITY [128 
 
 the previous reasoning to shew that every substitution, which 
 changes an a into a c, must change all the a's into c's. By 
 continuing this process, we finally divide the symbols into a 
 number of distinct sets of r each, such that every substitution of 
 the group must change the a's either into themselves or into 
 some other set : and therefore also must change every set either 
 into itself or into some other set. The group must therefore be 
 imprimitive. Hence : — 
 
 Theorem IV. If, among the symbols permuted by a transi- 
 tive group, it is possible to choose a set such that every 
 substitution of the group, which changes a chosen symbol of the 
 set either into itself or into another of the set, permutes all the 
 symbols of the set among themselves ; then the group is im- 
 primMive, and the set of symbols forms an imprimitive system. 
 
 Corollary I. I{ a^, a^, , a^ are a part of the symbols 
 
 permuted by a primitive group, there must be substitutions of 
 the group, which replace some of this set of symbols by others 
 of the set, and the remainder by symbols not belonging to the 
 set. 
 
 Corollary II. A sub-group of a primitive group, which 
 keeps one symbol unchanged, must contain substitutions which 
 displace any other symbol. 
 
 If the sub-group H, that leaves a^ unchanged, leaves every 
 
 symbol of the set aj, a^ osr unchanged, then H must be 
 
 transformed into itself by every substitution which changes any 
 one of these symbols into any other. Every substitution, which 
 changes one of the set into another, must therefore permute 
 the set among themselves ; and the group, contrary to supposi- 
 tion, is imprimitive. 
 
 129. It has already been seen that, in particular cases, it 
 may be possible to distribute the symbols, which are permuted by 
 an imprimitive group, into imprimitive systems in more than 
 one way. When this is possible, suppose that two systems 
 which contain Oj are 
 
 and ai, a2, , a,', a ^^i, , a^ ; 
 
129] IMPRIMITIVE SYSTEMS 185 
 
 and that the symbols common to the two systems are 
 
 A substitution of the group, which changes a^ into a',.+i, 
 
 must change o^, a^ , «,• into r symbols of that system of 
 
 the first set which contains a'^+i, while it changes the system of 
 the second set that contains a^ into itself. Hence the latter 
 system contains at least r symbols of that system of the first 
 set in which a'r+i occurs. By considering the effect of the 
 inverse substitution, it is clear that the system 
 
 cannot have more than r symbols in common with the system 
 of the first set that contains aV+i- Hence the n symbols of this 
 system can be divided into sets of r, such that each set is 
 contained in some system of the first set. It follows that n, and 
 therefore also m, must be divisible by r. 
 
 Suppose now that 61 is any symbol which is not contained 
 in either of the above systems. A substitution that changes 
 Oj into 61 must change the two systems into two others, which 
 have r symbols 
 
 ^1, h ,br 
 
 in common ; and since no two systems of either set have a 
 common symbol, these r symbols must be distinct from 
 
 Further, from the mode in which the set 61, 62. >K has 
 
 been obtained, any operation, which changes one of the symbols 
 
 ttj, a^, , a,, into one of the symbols 61, b.^, , b^, must 
 
 change all the symbols of the first set into those of the second. 
 Hence the symbols operated on by the group can be divided 
 into systems of r each, by taking together the sets of r 
 symbols which are common to the various pairs of the two 
 given sets of imprimitive systems ; and the group is imprimitive 
 in regard to this new set of systems of r symbols ' each. 
 Hence : — 
 
 Theorem V. If the symbols permuted by a transitive group 
 can be divided into imprimitive systems in two distinct ways, m 
 being the number of symbols in each system of one set and n in 
 
186 IMPRIMITIVE SYSTEMS [129 
 
 each system of the other ; and if some system of the first set has 
 r symbols in common with some system of the second set; then 
 (i) r is a factor of both m and n, a/)id (ii) the symbols can be 
 divided into a set of systems of r each, in respect of which the 
 group is imprimitive. 
 
 This result may also be regarded as an immediate conse- 
 quence of Theorem III, § 124. For if H^ and H2 are two 
 sub-groups of G each of which contains the sub-group R, and 
 if K is the greatest common sub-group of H^ and H^, then K 
 contains R. Now if R contains no self-conjugate sub-group 
 of G, then G can be represented as a transitive group whose 
 degree is the order of G divided by the order of R. If the 
 respective orders of Hi and IT^ are m times and n times the 
 order of R, the symbols can be divided in two distinct ways 
 into sets of imprimitive systems, the systems containing m 
 and n symbols respectively. Also, if the order of ^ is ?• times 
 the order of R, then r is a factor of m and of n ; by considering 
 the sub-group K, the symbols may be divided into a set of 
 systems which contain r symbols each. 
 
 It might be expected that, just as we can form a new set of 
 imprimitive systems by taking together the symbols which are 
 common to pairs of systems of two given sets, so we might form 
 another new set of systems by combining all the systems of one set 
 which have any symbols in common with a single system of the 
 other set. A very cursory consideration will shew however that 
 this is not in general the case. In fact, it is sufficient to point out 
 that, with the notation already used, the number of symbols in such 
 
 a new system would be - — ; and this number is not necessarily a 
 
 factor of the degree of the group. Also, even if this number is a 
 factor of the degree of the group, it will not in general be the case 
 that the symbols so grouped together form an imprimitive system. 
 
 130. We may now discuss, more fully than was possible in 
 § 106, the form of a self-conjugate sub-group of a given transitive 
 group. Such a sub-group must clearly contain one or more 
 operations displacing every symbol operated on by the group. 
 For if every operation of the sub-group keeps the symbol di 
 unchanged, then since it is self-conjugate, every operation will 
 
 keep 0^2, Kg, , unchanged: and the sub-group must reduce 
 
 to the identical operation only. 
 
130] SELF-CONJUGATE SUB-GROUPS OF TRANSITIVE GROUPS 187 
 
 Suppose now, if possible, that H is an intransitive self- 
 conjugate sub-group of a transitive group G ; and that H 
 permutes the n symbols of G in the separate transitive sets 
 
 01,02, ,a,n,;h,h, ,bn^; If /S is any operation 
 
 of G which changes a^ into h^, then, since 
 
 it must change all the a's into h's ; and since 
 
 S~^ must change all the b's into a's. Hence the number of 
 symbols in the two sets, and therefore the number of symbols 
 in each of the sets, must be the same. 
 
 Moreover every operation of G, since it transforms H into 
 itself, must either permute the symbols of any set among 
 themselves, or it must change them all into the symbols of 
 some other set. Hence G must be imprimitive, and H must 
 consist of those operations of G which permute the symbols of 
 each imprimitive system among themselves. 
 
 Conversely, when G is imprimitive, it is immediately obvious 
 that those operations of G, if any such exist, which permute the 
 symbols of each of a set of imprimitive systems among them- 
 selves, form a self-conjugate sub-group. Hence : — ■ 
 
 Theorem VI. A self-conjugate sub-group of a primitive 
 group must be transitive ; and if an imprimitive group has an 
 intransitive self-conjugate sub-group, it must consist of the 
 operations which permute among themselves the symbols of each 
 of a set of imprimitive systems. 
 
 If G is an imprimitive group of degree mn, and if there are 
 n imprimitive systems of m symbols each, then we have seen 
 in § 123 that G is isomorphic with a group G' of degree n. In 
 particular instances, it may at once be evident, from the order of 
 G, that this isomorphism cannot be simple. For example, if 
 the order of G has a factor which does not divide n !, this is 
 certainly the case : and more generally, if it is known indepen- 
 dently that G cannot be expressed as a transitive group of 
 degree n, then G must certainly be multiply isomorphic with 
 G'- In such instances the self-conjugate sub-group of G, which 
 
188 SELF-CONJUGATE SUB-GROUPS [130 
 
 corresponds to the identical operation of 0', is that intransitive 
 self-conjugate sub-group, which interchanges among themselves 
 the symbols of each imprimitive system. 
 
 If G is soluble, a minimum self-conjugate sub-group of G 
 must have for its order a power of a prime. Also, if G has an 
 intransitive self-conjugate sub-group, it must have an intransi- 
 tive minimum self-conjugate sub-group. Hence if G is soluble 
 and has intransitive self-conjugate sub-groups, the symbols 
 permuted by G must be capable of division into imprimitive 
 systems, such that the number in each system is the power of 
 a prime. 
 
 131. Let (r be a A-ply transitive group of degree n{k> 2), 
 and let G,- be that sub-group of G which keeps r{<k) given 
 symbols unchanged, so that Gr is (A; — r)-ply transitive in the 
 remaining n — r symbols. Also, let H he a, self-conjugate 
 sub-group of G, and let Hr be that sub-group of H which keeps 
 the same r symbols unchanged ; so that Hr is the common 
 sub-group of H and (?,.. Since every operation of Gr transforms 
 both H and Gr into themselves, every operation of Gr must be 
 permutable with Hr ; i.e. Hr is a self-conjugate sub-group of &,.. 
 Now, if r = k — 2, Gjc^ is doubly transitive in the n—k+2 
 symbols on which it operates ; it is therefore primitive. Hence, 
 unless Hje^ consists of the identical operation only, it must be 
 transitive in the n — k+2 symbols. 
 
 If H;c-i is the identical operation, H contains no operation, 
 except identity, which displaces less than n — k+S symbols. 
 Suppose, first, that H contains operations, other than identity, 
 which leave one or more symbols unchanged. Then, since H 
 is a self-conjugate sub-group and G is A;-ply transitive, it may 
 be shewn, exactly as in § 110, that H must contain operations 
 displacing not more than 2k— 2 symbols. Hence Hje-^ can 
 consist of the identical operation alone, only if 
 
 n-k + Si!^2k-2, 
 or A -"t Jn + f . 
 
 When this inequality holds, we have seen (p. 152) that G 
 contains the alternating group. Hence in this case, if G does 
 not contain the alternating group, it follows that Hie-2 is transi- 
 tive in the n — k + 2 symbols on which it operates. 
 
131] OF PRIMITIVE GROUPS 189 
 
 Since H is self-conjiigate and G is A;-ply transitive, H must 
 contain a sub-group conjugate to Hjc—^ which keeps any other 
 A; — 2 symbols unchanged. Hence J^^-s must be doubly transi- 
 tive in the w — A; + 3 symbols on which it operates ; and so on. 
 Finally, if is not the symmetric group (the alternating group, 
 being simple, contains no self-conjugate sub-group) H must be 
 {k — l)-ply transitive. 
 
 Suppose, next, that H contains no operation, except identity, 
 which leaves any symbol unchanged. Then if, with the notation 
 of § \\Q,j = k — l for every operation of H, the argument there 
 used does not apply. For it is impossible to choose the operation 
 T so that Ci is a symbol which does not occur in S. 
 
 The self-conjugate sub-group H contains a single operation 
 changing a given symbol a^ into any other symbol a^. Also G 
 contains operations which leave a^ unchanged and change a,, 
 into any other symbol ag. Hence the operations of H, other 
 than identity, form a single conjugate set in G; and therefore 
 
 H must be an Abelian group of order p^ and type (1,1, , to 
 
 m units); p being a prime. Further, since G is by supposition 
 at least triply transitive, it must contain operations which 
 transform any two operations of H, other than identity, into 
 any other two. If p were an odd prime, and Pi and P^ were 
 two of the generating operations of H, it follows that G would 
 have an operation S such that 
 
 S-^P^S = P, , 8-'P,S = Pi" ; 
 and this is impossible. Hence p must be 2. Further if Q were 
 more than triply transitive, and if ^, B, were three independent 
 generating operations of H, then G would have an operation S 
 such that 
 
 t-'A-Z = A, t-^Bt = B, %-'GX = AB. 
 This again is impossible, and therefore k must be 3. Hence : — 
 
 Theorem VII. A self-conjugate sub-group of a k-ply 
 transitive group of degree n (2<k<n), is in general at least 
 {k—l)-ply transitive^. The only exception is that a triply 
 transitive group of degree 2"* may have a self-conjugate sub- 
 group of order 2"*. 
 
 ^ Jordan, Traite des Substitutions, p. 65 ; where, however, the exceptional 
 ease is overlooked. 
 
190 MINIMUM SELF-CONJUGATE SUB-GROUPS [132 
 
 132. For the further discussion of the self-conjugate 
 sub-groups of a primitive group, it is necessary to consider in 
 what forms the direct product of two groups can be represented 
 as a transitive group. 
 
 Let G be the direct product of two groups H^ and H^, and 
 suppose that Q can be represented as a transitive group 
 of degree n. When G is thus represented, we will suppose 
 that Hi is transitive in the n symbols that Q permutes. We 
 have seen in § 107 that every substitution of n symbols, which 
 is permutable with each of the substitutions of a group 
 transitive in the n symbols, must displace all the n symbols. 
 It follows that every substitution of H^ must displace all the n 
 symbols on which G operates; and that the order of H^ is 
 equal to or is a factor of n. 
 
 If the order of H^ is equal to n, then H.^ is transitive in the 
 n symbols, so that the order of H^ cannot be greater than n. 
 In this case, H^ and H^. must (§ 107) be two simply isomorphic 
 groups of order n, which have no self-conjugate operations 
 except identity. Further, if H-^ and H^ in this case are not 
 simple groups, let .ST be a self-conjugate sub-group of H^. 
 Since every operation of H^ is permutable with every operation 
 of ifi , ilT is a self-conjugate sub-group of G. Now the order of 
 K is less than n, the degree of G ; therefore K is intransitive 
 and G is imprimitive. On the other hand, we have seen 
 {loc. cit.) that, if H^ and H^ are simple, the sub-group of 
 that keeps one symbol iixed is a maximum sub-group : and 
 therefore G is primitive. Hence : — 
 
 Theorem VIII. // the direct product V of Hi and H^ can 
 he represented as a transitive group of degree n, in such a way 
 that Hi and H2 are transitive sub-groups of F, then H^ and H^ 
 must he simply isomorphic groups of order n, which have no 
 self -conjugate operations except identity. When this condition 
 is satisfied, T will he primitive if, and only if, H^ and H^ are 
 simple. 
 
 133. Suppose now that a primitive group G, of degree n, 
 has two distinct minimum self-conjugate sub-groups H^ and 
 H,. Then every operation of H^ (or H^) is permutable with 
 
134] OF PRIMITIVE GROUPS 191 
 
 H^ (or jETi), and Hi, H^ have no common operation except 
 identity. Hence (§ 34) the group [H,, H^, which we will call 
 r, is the direct product of H^ and Ho^. Now T is a self- 
 conjugate sub-group of G: it is therefore transitive in 
 the n symbols which Q permutes. Also H^ and H^, being 
 self-conjugate sub-groups of 0, are transitive. Hence, by 
 Theorem VIII (§ 132), H^ and H^ are simply isomorphic, and 
 n is equal to the order of H^. Moreover, since H^ is a minimum 
 self-conjugate sub-group of G which contains no self-conjugate 
 operations except identity, it must (Theorem IV, § 94) be either 
 a simple group of composite order, or the direct product of 
 several simply isomorphic simple groups of composite order. It 
 follows that G cannot have two distinct minimum self-conjugate 
 sub-groups unless the degree of G is equal to or is a power 
 of the order of some simple group of composite order. 
 
 134. Let now F be a minimum self-conjugate sub-group 
 of a doubly transitive group G, and suppose that F is the direct 
 product of the a simply isomorphic simple groups 
 
 -ffl, ^2, , Ha.. 
 
 Since G is primitive, F is transitive. If Hi is a cyclical 
 group of prime order p, the order of F is p" ; therefore the 
 degree of F, or what is the same thing, the degree of G, is p". 
 
 If Hi is a simple group of composite order, and if a > 2, 
 then (§ 132) Hi cannot be transitive. The intransitive systems 
 of Hi, since they form a set of imprimitive systems for F, must 
 each contain the same number m of symbols. If m is less than 
 the order ot Hi, a. sub-group of Hi which leaves unchanged one 
 symbol of -one intransitive system will leave unchanged one 
 symbol of each intransitive system. Now we shall see, in § 136, 
 that the operations of an imprimitive self-conjugate sub-group 
 of a doubly transitive group must displace all or all but one of 
 the symbols. Hence m cannot be less than the order of H. 
 We may similarly shew that, if m is equal to the order of Hi, 
 some of the operations of F must keep more than one symbol 
 fixed ; and therefore, if a > 2, the group assumed cannot exist. 
 If a = 2, Hi may be transitive. But in this case {H, H^} 
 certainly contains operations which leave more than one symbol 
 unchanged ; and again the group assumed cannot exist. Hence 
 
192 SOLUBLE PRIMITIVE GROUPS [134 
 
 finally no doubly transitive group can contain a minimum self- 
 conjugate sub-group of the type assumed. 
 
 No general law can be stated regarding self-conjugate 
 sub-groups of simply transitive primitive groups; but for 
 groups which are at least doubly transitive the preceding 
 results may be summed up as follows: — 
 
 Theorem IX. A group which is at least doubly transitive 
 either must be simple or must contain a simple group H as a 
 self-conjugate sub-group. In the latter case no operation of G, 
 except identity, is permutable with every operation of H. The 
 only exceptions to this statement are that a triply transitive group 
 of degree 2" may have a self-conjugate sub-group of order 2™; 
 and that a doubly transitive group of degree p^, where p is a 
 prime, may have a self-conjugate sub-group of order p™ 
 
 CoroUary. If a primitive group is soluble, its degree 
 must be the power of a prime^. 
 
 In fact, if a group is soluble, so also is its minimum 
 self- conjugate sub-group. The latter must be therefore an 
 Abelian group of order jp": and since this group must be 
 transitive, its order is equal to the degree of the primitive 
 group. 
 
 135. As illustrating the occurrence of an imprimitive self- 
 conjugate sub-group in a primitive group, we will construct a 
 primitive group of degree 36 which has an imprimitive self-conju- 
 gate sub-group. For this purpose, let^ 
 ^ = (1, 2, 3) (4, 5, 6) (7, 8, 9) (10, 11, 12) (13, 14, 15) 
 
 (16, 17, 18) (19, 20, 21) (22, 23, 24) (25, 26, 27) 
 
 (28, 29, 30) (31, 32, 33) (34, 35, 36), 
 
 and A = {3, 4) (5, 6) (9, 10) (11, 12) (15, 16) (17, 18) 
 
 (21, 22) (23, 24) (27, 28) (29, 30) (33, 34) (35, 36); 
 so that {S, A} is an intransitive group of degree 36, the symbols 
 being interchanged in 6 transitive systems containing 6 symbols 
 each. This group is simply isomorphic with 
 {(123) (456), (34) (56)}; 
 
 1 This result, stated in a somewhat different form, is given, among many 
 others, in the letter written by Galois to his friend Chevalier on the evening 
 of May 29th, 1832, the day before the duel in which he was killed. The letter 
 was first printed in the Bevue Encyclopidique (1832), p. 568 ; it was reprinted 
 in the collection of Galois's mathematical writings in Liouville's Journal, t. xi 
 (1846), pp. 381—444. 
 
 ^ The commas in the symbols for the substitutions are here used to prevent 
 confusion among the one-digit and two-digit numbers. 
 
136] EXAMPLES 193 
 
 and it may be easily verified that this group is simply isomorphic with 
 the alternating group of 5 symbols, the order of which is 60. 
 
 Also let 
 /=(2,7)(3, 13) (4, 19) (5, 25) (6, 31) (9, 14) (10, 20) (11, 26) 
 
 (12, 32) (16, 21) (17, 27) (18, 33) (23, 28) (24, 34) (30, 35). 
 Then 
 JSJ={1, 7, 13) (19, 25, 31) (2, 8, 14) (20, 26, 32) 
 
 (3, 9, 15) (21, 27, 33) (4, 10, 16) (22, 28, 34) 
 
 (5, 11, 17) (23, 29, 35) (6, 12, 18) (24, 30, 36), 
 and JAJ=(U, 19) (25, 31) (14, 20) (26, 32) (15, 21) (27, 33) 
 
 (16, 22) (28, 34) (17, 23) (29, 35) (18, 24) (30, 36) ; 
 
 and {JSJ, JAJ] is similar to and simply isomorphic with {S, A}. 
 
 Now it may be directly verified that S and A are, each of them, 
 permutable with JSJ and JAJ ; and therefore every operation of 
 the group {S, A] is permutable with every operation of the group 
 \JSJ, JAJ}. Also these two groups can have no common operation, 
 since the symbols into which {.S', A] changes any given symbol are 
 all distinct from those into which {JSJ, JAJ] change it. Hence 
 [S, A, JSJ, JAJ} is the direct product of {S, A} and {JSJ, JAJ}; 
 it is therefore a group of order 3600. It is also, from its mode of 
 formation, a transitive group of degree 36 ; and it interchanges the 
 symbols in two and only two distinct sets of imprimitive systems, of 
 which 
 
 1, 2, 3, 4, 5, 6 and 1, 7, 13, 19, 25, 31 
 
 may be taken as representatives. 
 
 Now J does not interchange the 36 symbols in either of these 
 systems, and therefore it cannot occur in {S, A, JSJ, JAJ}. 
 Further 
 
 J{S, A, JSJ, JAJ} J= {S, A, JSJ, JAJ} ; 
 
 and therefore {/, >S', A} is a transitive group of degree 36 and order 
 7200. Also, since J does not interchange the symbols in either of 
 the two sets of imprimitive systems of {S, A, JSJ, JAJ}, it follows 
 that {J, S, A} is primitive. 
 
 136. We have seen that, with a single exception, a A-ply 
 transitive group {k > 2) cannot have an imprimitive self-conjugate 
 sub-group ; while the example in the preceding section illustrates 
 the occurrence of an imprimitive self-conjugate sub-group in a simply 
 transitive primitive group. We wdl now consider, from a rather 
 different point of view, the possibility of an imprimitive self- 
 conjugate sub-group in a doubly transitive group. Let (? be a 
 doubly transitive group of degree mn, and let S be an imprimitive 
 self-conjugate sub-group of G. Suppose that m is the smallest 
 number, other than unity, of symbols which occur in an imprimitive 
 system ; and let 
 
 (I'll ^2) ) '^m 
 
 B. 13 
 
194 EXAMPLES [136 
 
 form an imprimitive system. Since G is doubly transitive, it must 
 contain a substitution S, which leaves % unchanged and changes Oj 
 into «m+i, a symbol not contained in the given set. If .S" changes 
 the given set into 
 
 then, since 
 
 this new set must form an imprimitive system for H. Also, since 
 m is the smallest number of symbols that can occur in an imprimitive 
 system, the two sets have no symbol in common except a^. 
 
 Now a^^i may be any symbol not contained in the original 
 system. Hence it must be possible to distribute the mn symbols 
 into sets of imprimitive systems of m each, such that every pair of 
 symbols occurs in one system and no pair in more than one system. 
 This implies that mn—1 is divisible bym— 1, or that m — 1 is 
 divisible by m— 1. 
 
 Consider now a substitution of H which leaves a^ unchanged. 
 It must permute among themselves the remaining m - 1 symbols 
 
 of each of the r- systems in which a^ occurs. If a^ is any 
 
 other symbol, a similar statement applies to it. Now no two 
 systems have more than one symbol in common. Hence every 
 substitution, which leaves both Oj and a^ unchanged, must leave all 
 the symbols unchanged. The sub-group H is therefore such that 
 each of its substitutions displaces all or all but one of the symbols. 
 Moreover the substitutions, which leave a^ unchanged, permute 
 among themselves the remaining symbols of each system in which 
 «! occurs; therefore the order of H must be niw/t, where /«. is a 
 factor of m — \. 
 
 It has been seen in § 1 1 2 that, for certain values of mn, groups 
 satisfying these conditions actually exist. The doubly transitive 
 groups, of degree p™ and order p™(^™-l) there obtained, have 
 imprimitive self-conjugate sub-groups of orders ^™ (p - 1) and p™- 
 
 137. Ex. 1. Shew that a transitive group of order M and 
 degree 2ra, which is imprimitive in respect of two systems of n 
 symbols each, must have a self-conjugate sub-group of order ^N. 
 
 Ex. 2. Shew that an imprimitive group of order Fp^ and 
 degree np, where p is an odd prime, which has p imprimitive 
 systems of n symbols each, must contain a self-conjugate sub-group 
 whose order is a multiple of p. 
 
 Ex. 3. Shew that, if n is the smallest number of symbols in 
 which a group G can be represented as a transitive group, then n" is 
 the smallest number of symbols in which the direct product of a 
 groups, simply isomorphic with G, can be represented as a transitive 
 group. 
 
137] EXAMPLES 195 
 
 Ex. 4. Prove that if a group G, of order N, represented as a 
 transitive substitution group, is imprimitive only when its degree is 
 N ; then either G is AbeHan, or N is the product of two distinct 
 primes. (Dyck.) 
 
 Ex. 5. Shew that if, in the tableau of p. 174, the sub-group 
 
 1, O2, O3, , Ofi 
 
 is a self- conjugate sub-group H of the given group G, then, in each 
 square compartment of the tableau, every horizontal line contains 
 the same /u, symbols. 
 
 Shew also that, if each square compartment of the tableau is 
 regarded as a single symbol, the permutations of these symbols 
 
 given by the tableau represents the group ^ in regular form. 
 
 (Dyck.) 
 
 13—2 
 
CHAPTER X. 
 
 ON SUBSTITUTION GROUPS : TRANSITIVITY AND 
 PRIMITIVITY : (concluding PROPERTIES.) 
 
 138. The determination of all distinct transitive sub- 
 groups of the symmetric group of n symbols has been carried 
 out for values of n up to 12*- This investigation, if carried out 
 for sufficiently great values of n, would involve the expression 
 of all types of group in all possible transitive forms. 
 
 From the point of view of one of the chief problems of pure 
 group-theory, namely, the determination of all distinct types 
 of group of a given order, this analysis of the symmetric group 
 of n symbols is not a succinct process, as it continually involves 
 the redetermination of groups which have been already ob- 
 tained. Thus a simple group of degree mn, in which the symbols 
 are permuted in m imprimitive systems of n each, would in 
 this analysis have been already obtained as a group of degree m. 
 With reference then to the more restricted problem of de- 
 termining all types of simple groups, it would certainly be 
 sufficient to find all primitive sub-groups of the symmetric 
 group. 
 
 139. We shall proceed to determine a superior limit to the 
 order of a primitive group of degree n, other than the alter- 
 nating or the symmetric group. 
 
 * Jordan, Comptes Rendm, t. lxxiii (1871), pp. 853—857; ib. t. lxxv 
 (1872), pp. 1754—1757. 
 
 And Bee the Bulletin of the New York Mathematical Society, Cole, 1st series, 
 Vol. II, pp. 184—190; 250—258: Miller, lat series. Vol. iii, pp. 168, 169; 242— 
 245 : 2nd series. Vol. i, pp. 67—72; 255—258 : Vol. ii, pp. 138—145. Quarterly 
 Journal of Mathematics, Cole, Vol. xxvi, pp. 372—388: Vol. xxvn, pp. 35—50: 
 Miller, Vol. xxvii, pp. 99—118; Vol. xxviii, pp. 193—231. These memoirs 
 ■were all published between 1892 and 1896 ; in them further references will be 
 found. 
 
140] PRIMITIVE GROUPS 19^ 
 
 Let be a primitive group of degree ;?, and suppose that G 
 contains a sub-group H which leaves n - m symbols unchanged 
 and is transitive in the remaining m. Since G is primitive, H 
 and the sub-groups conjugate to it must generate a transitive 
 self-conjugate sub-group of G; and therefore there must be 
 some sub-group H', conjugate to H, such that the m symbols 
 operated on by H and the m operated on by H' are not all 
 distinct. Suppose H' is chosen so that these two sets of m 
 symbols have as great a number in common as possible, say s ; 
 and represent by 
 
 «!, aj, , a,., 7i, 73, , 7s, 
 
 and /3i, ^a, ^<-> 7i> 72> 7s) 
 
 where r + s = m, the symbols operated on by H and H' respec- 
 tively. Then {H, H'] is a transitive group in the 2r -H s symbols 
 a, /S, and 7, which leaves unaltered all the remaining symbols 
 of G. 
 
 If t)! is an operation of H which changes Ha into a^, S~^H'S 
 does not aftect a^. Hence, unless S interchanges the a's among 
 themselves, the m symbols operated on by H' and the m 
 operated on by S~^II'8 will have more than s in common. 
 Every operation of H which changes one a into another must 
 therefore interchange all the a's among themselves; hence H 
 must be imprimitive. 
 
 If then H is primitive, s must be equal to m — 1. In any 
 ease, if s = m - 1, {H, H'} is a doubly transitive and therefore 
 primitive group of degree m + 1, which leaves the remaining 
 » — m — 1 symbols of G unchanged. We may reason about this 
 sub-group as we have done about H. Among the sub-groups 
 conjugate to {H, B'}, there must be one at least which operates 
 on m of the symbols displaced by {H, H'}. This, with {H, H'], 
 generates a triply transitive group of degree m + 2, which 
 leaves n — m — '2 symbols imchanged. Proceeding thus, we find 
 finally that G itself must be {n — m+ l)-ply transitive. 
 
 140. If s is less than ?)i— 1, we may again deal with the 
 sub-group {H, H'}, or H^, exactly as we have dealt with H. It 
 is a transitive group of degree m^ (> Hi), which leaves n — m^ 
 symbols unchanged. If, among the sub-groups conjugate to 
 
198 PRIMITIVE [14,0 
 
 Hi, none operates on more than s^ of the symbols affected by 
 Hi, and if H^ is a suitably chosen conjugate sub-group, then 
 {Hi, Hi] is a transitive group of degree 2mi — Si, which leaves 
 n — 2«ii + Si symbols unchanged. Continuing this process, we 
 must, before arriving at a group of degree n, reach a stage at 
 which the number s,. is equal to m,. — 1. 
 
 For suppose, if possible, that among the groups conjugate 
 to K, of degree p + o", none displaces more than o- of the 
 symbols acted on by K, while at the same time 2p + a=n. If 
 
 «!. aj, , Hp, 7i, 72 , 7cr, 
 
 and I3i, A, /8p- 7i. 7-2. 7<r 
 
 are the symbols affected by K and K' respectively, then since G is 
 primitive, it must contain an operation 8 which changes Oj into 
 tta without at the same time changing all the a's into a's. If 
 then we transform K' by S, the two groups K' and S~^K'8 must 
 operate on more than a common symbols, contrary to supposi- 
 tion. 
 
 Hence G must in this case certainly contain a transitive 
 sub-group of degree n—1, and therefore is itself at least 
 doubly transitive \ 
 
 141. Returning to the case in which H is primitive and G 
 therefore (m — m+ l)-ply transitive, we at once obtain an 
 inferior limit for m. We have seen, in fact, in Theorem IV, 
 § 110, that a group of degree n, other than the alternating or 
 the symmetric group, cannot be more than (^n + l)-ply tran- 
 sitive. Hence 
 
 n — m+1 'Jf-^n + l, 
 or m-^ |n. 
 
 We may sum up these results as follows : — 
 
 1 The results contained in §§ 139, 140 are due to Jordan (Liouville's Journal, 
 Vol. XVI, 1871) and Netto {Crelle's Journal, Vol. era, 1889). They have been 
 extended by Marggraff: "Ueber primitiven Gruppen mit transitiven Unter- 
 gruppen geringeren Grades," {Inaugural Dissertation, Giessen, 1892). With the 
 notation used in the text, MarggrafiE shews that, unless the symbols affected by 
 H can be divided into imprimitive systems of r symbols each, in at least r + 1 
 distinct ways, O will be (re-m + l)-ply transitive. In particular, if H is a 
 cyclical group of degree m, G is (n-?re + l)-ply transitive. He also shews that 
 in any case m ^ J«. 
 
142] GROUPS 199 
 
 Theorem I. A primitive group G of degree n, which has a 
 sub-group H that keeps n — m symbols unchanged and is transi- 
 tive in the remaining m symbols, is at least doubly transitive. If 
 H is primitive and does not contain the alternating group, m 
 cannot be less than |w, and is (n — m + V)-ply transitive. 
 
 Corollary. The order of a primitive group of degree n 
 
 n\ 
 
 cannot exceed -jr— j; , where 2, 3, « are the distinct 
 
 2.3 p ^ 
 
 primes which are less than §n.. 
 
 If §" is the highest power of a prime q that divides n !, the 
 sub-groups of order g" of the symmetric group form a single 
 conjugate set, and each of them must contain circular substitu- 
 tions of order q. Hence if g' < |n, it follows by the theorem 
 that no primitive group of degree n, other than the alternating 
 or the symmetric group, can contain a sub-group of order g" ; 
 and therefore g""^ is the highest power of q that can divide 
 the order of the group. 
 
 142. The ratio of 2.3 p to n increases rapidly as n 
 
 increases, and it is at once obvious that, when n > 7, this ratio is 
 greater than unity; hence for values of n greater than 7, the 
 symmetric group can have no primitive sub-group of order 
 {n-l)\. 
 
 The order of the greatest imprimitive sub-group of the 
 
 fn \* 
 symmetric group is a! f-!j , where a is the smallest factor of 
 
 n. When n > 4, this is less than (n — 1) !. 
 
 The order of the greatest intransitive sub-group of the 
 symmetric group, other than the sub-groups that keep one 
 symbol fixed, is 2 ! (n. - 2)!. This is always less than {n-l)\. 
 
 Hence when n>7, the only sub-groups of order {n-l)\ of 
 the symmetric group are the sub-groups which each keep one 
 symbol fixed ; and these form a conjugate set of n sub-groups. 
 
 When n, = 7, a sub-group of order (?i - 1)! must be intransi- 
 tive, and therefore the same result holds in this case ; this also 
 is true when n is 3, 4, or 5. 
 
200 PRIMITIVE [142 
 
 Lastly, when n=Q, there may, by the foregoing theorem, 
 be primitive sub-groups of order 6!. That such sub-groups 
 actually exist may be verified at once by considering the 
 symmetric group of 5 symbols. This group contains 6 cyclical 
 sub-groups of order 5, and each of them is self-conjugate in a 
 sub-group of order 20. Hence, since the only self-conjugate 
 sub-group contained in the symmetric group of degree 5 is the 
 corresponding alternating group, the symmetric group of degree 
 5 can be expressed as a doubly transitive group of degree 6. 
 The symmetric group of degree 6 therefore contains a set of 6 
 conjugate doubly transitive sub-groups of order 5!, which are 
 simply isomorphic with the intransitive sub-groups that each 
 keep one symbol fixed. Finally, if the 12 sub-groups of order 
 5!, which are thus accounted for, do not exhaust all the sub- 
 groups of this order, any other would have in common with 
 each of the 12 a sub-group of order 20; and therefore the 
 operations of order 3 contained in it would be distinct from 
 those in the previous 12. But these clearly contain all the 
 operations of order 3 of the symmetric group, and therefore 
 there can be no other sub-groups of order 5!. Hence: — 
 
 Theorem II. The symmetric group of degree n (ra 4= 6) 
 contains n and only n sub-groups of order (n — 1)!, which form a 
 single conjugate set. The symmetric group of degree 6 contains 
 12 sub-groups of order 5!, which are simply isomorphic with one 
 another and form two conjugate sets of 6 each. 
 
 143. We shall now discuss certain further limitations on 
 the order of a primitive group of given degree. Though it will 
 be seen that these do not lead to general results, similar to that 
 given by Theorem I, § 141, yet in many special cases they are 
 of considerable assistance in determining the possible existence 
 of groups of given orders and degrees. 
 
 We consider first a group G of order N and of prime degree jp. 
 If G is not cyclical, it must contain substitutions which keep 
 only one symbol unchanged. For let P be a substitution of G of 
 order p. The only substitutions permutable with P are its own 
 powers (§ 107) ; and the only substitutions permutable mth (P| 
 are substitutions which keep one symbol unchanged and are regu- 
 
143] GROUPS 201 
 
 lar in the remaining p—1 symbols (§ 112)* Now if the only sub- 
 
 iV 
 stitutions permutable with {P} are its own, then {P} is one of — 
 
 conjugate sub-groups; and these contain Nil J substitu- 
 
 tions of order p. In this case, G would contain exactly — 
 
 substitutions whose orders are not divisible by p. But this is 
 
 clearly impossible, since — is the order of a sub-group which 
 
 keeps one symbol unchanged, and there are p such sub-groups. 
 Hence there must be substitutions in G, other than those of 
 {P\, which are permutable with [P] ; and each of these substi- 
 tutions keeps one symbol unchanged. 
 
 It follows from § 134 that G, if soluble, must contain a 
 self-conjugate sub-group of order p : therefore no group of 
 prime degree p, which contains more than one sub-gi'oup of 
 order p, can be soluble. 
 
 li 1+kp is the number of sub-groups of order p contained 
 in G, then 
 
 F^pP^(l + kp), 
 
 where d is a factor oi p—1; and a sub-group of order p is 
 transformed into itself by every substitution of a cyclical 
 
 p — 1 
 
 sub-group of order ^~~ . When d is odd, a substitution which 
 
 generates this cyclical sub-group is an odd substitution; and 
 G then contains a self-conjugate sub-group of order ^JSf. 
 
 If both p and |(p — 1) are primes, the order of a group of 
 degree p, which contains more than one sub-group of order p, 
 must be divisible by ^ (^ - 1). For if the order is not divisible 
 ^y Hp- 1), the order of the sub-group, within which a sub- 
 group of order p is self-conjugate, must be 2p. Now the 
 
 * It is shewn in § 112 that {P} is permutable with a circular substitution of 
 ^ - 1 symbols, which leaves one symbol Oj unchanged. If there are other sub- 
 stitutions which leave a^ unchanged and are permutable with {P}, some such 
 substitution will leave two symbols unchanged. This is clearly impossible. 
 Hence the group of order ^ (p - 1) is the greatest group of the p symbols in which 
 {P\ is self-conjugate. 
 
202 PRIMITIVE GROUPS [143 
 
 substitutions of order 2 in this sub-group consist of ^(«_i) 
 transpositions, so that they are odd substitutions. The group 
 must therefore contain a self-conjugate sub-group in which 
 these operations of order 2 do not occur. In such a sub-group, 
 the only operations permutable with those of a sub-group of 
 order p are its own ; and we have seen that no such group can 
 exist. The order of the group must therefore, as stated above, 
 be divisible hy \{p-V). 
 
 144. Let G' be a primitive group of degree n and order N; 
 and let p be a prime, which is a factor of N but not of either n 
 or ?i - 1. Moreover, suppose that n is congruent to v, mod. p ; 
 V being less than p. lin< p^ and if p« is the highest power of 
 p which divides N, the sub-groups of order p"^ must be Abelian 
 groups of type (1, 1, ... to a units). In fact, such a sub-group 
 must be intransitive, and, since n < p\ the number of symbols 
 in each transitive system of the sub-group must be p. In 
 any case the number of symbols left unchanged by a sub-group 
 of order ^" is of the form kp + v. 
 
 iV 
 Suppose now that, in a sub-group of order — which leaves 
 
 one symbol unchanged, a sub-group H of order p'^ is one of 
 
 — conjugate sub-groups. Then each of the n sub-groups 
 
 JSf 
 that keep one symbol unchanged contains — ^ — sub-groups of 
 
 order p°'; and each sub-group of order jp" belongs to kp + v 
 
 sub-groups that keep one symbol unchanged. Hence G 
 
 N 
 
 contains t-. r sub-groups of order «", and any one of 
 
 p«m (kp + v) ° '^ . ^ ' ■' 
 
 them, say H, is contained self-conjugately in a sub-group / of 
 order _p"m (kp + v). This sub-group / must interchange transit- 
 ively among themselves the kp + v symbols left unchanged by 
 H. For let a and h be any two of these symbols ; and let 8 he 
 an operation which changes a into h and transforms H into H'. 
 There must be an operation T which keeps h unchanged and 
 transforms H' into H, since in the sub-group that keeps 6 
 unchanged there is only one conjugate set of sub-groups of 
 order p" Then ST changes a into h and transforms H into 
 
145] 
 
 EXAMPLES 
 
 203 
 
 itself; and therefore / contains substitutions which change a 
 into h. Now it may happen that the existence of a sub-group 
 such as / requires that Q is either the alternating or the sym- 
 metric group. When this is the case, we infer that the order 
 of a primitive group of degree n, other than the alternating or 
 the symmetric group, caimot be divisible by p. 
 
 145. As a simple example, we will shew that the order of 
 a group of degree 19 cannot be divisible by 7, unless it contains 
 the alternating group. It follows from Theorem I, Corollary, 
 § 141, that the order of a group of degree 19, which does not 
 contain the alternating group, cannot be divisible by a power of 7 
 higher than the first, and that if the group contains a substitution 
 of order 7, the substitution must consist of two cycles of 7 symbols 
 each. The sub-group of order 7 must therefore leave 5 symbols 
 unchanged ; hence, by § 144, it must be contained self-conjugately in 
 a sub-group whose order is divisible by 5. ISTow (§ 83) a group of 
 order 35 is necessarily Abelian ; so that the group of degree 19 must 
 contain a substitution of order 5 which is permutable with a sub- 
 stitution of order 7. Such a substitution of order 5 must clearly 
 consist of a single cycle, and its presence in a group of degree 19 
 requires that the latter should contain the alternating group. It 
 follows that, if a group of degree 19 does not contain the 
 alternating group, its order is not divisible by 7. 
 
 As a second example, we will determine the possible forms for 
 the order of a group of degree 1 3, which does not contain the alter- 
 nating group. It follows, from Theorem I, Corollary, § 141, that 
 the order of such a group must be of the form 2" . 3^ . 5''' . 11 . 13 ; 
 where a, /?, y, S do not exceed 9, 4, 1, 1 respectively. 
 
 Suppose, first, that y is unity, if possible. A substitution of 
 order 5 must consist of two cycles of 5 symbols each; and a sub- 
 group of order 5 must therefore be self -conjugate in a sub-group of 
 order 15. There is then a substitution of order 3 which is per- 
 mutable with a substitution of order 5. Such a substitution must, 
 as in the last example, consist of a single cycle; and its existence 
 would imply that the group contains the alternating group. It 
 follows that, for the group as specified, y must be zero. 
 
 Suppose, next, that 8 is unity, if possible. The group is then 
 (Theorem I, § 141) triply transitive ; and the order of the sub-group, 
 that keeps two symbols fixed and is transitive in the remaining 11, 
 is 2". 3^. 11 ; and this sub-group must contain more than one sub- 
 group of order 11. We have seen in § 143 that no such group can 
 exist. Therefore 8 must be zero. 
 
 The two smallest numbers of the form 2™ 3" which are congruent 
 to unity, mod. 13, are 3^ and 2*3^; and every number of this form, 
 
204 PRIMITIVE GROUPS [145 
 
 which is congruent to unity, mod. 13, can be written (2^3^)'"3*. 
 Hence the order of every group of order 13, which contains no odd 
 substitution, must be of the form (2'' 3^)^. 3^''.». 13, where 2 is 2, 3 
 or 6. Since 3* is the highest power of 3 that can divide the order 
 of the group, the only admissible values of x and y are (i) x = 0, 
 2/ = 1 : (ii) X-—2, y = : (iii) « = 1, y = 0. 
 
 Suppose, first, that x = 0, y = l. The order of the group is 
 2 . 3M3, 3M3, or 2 . 3M3. There must be 13 sub-groups of order 
 3^* (or 3^), and since 13 is not congruent to unity, mod. 9, there must 
 be sub-groups of order 3' (or 3^) common to some two sub-groups of 
 order 3* (or 3'). Such a sub-group must be self -conjugate (Theorem 
 III, § 80). This case therefore cannot occur. 
 
 Next, suppose that x=2, y = 0. Then z must be 2, and the 
 order of the group is 2' . 3^ . 13. Now it is easy to verify that 2'. 3' 
 is not a possible order either for an intransitive or for an imprimitive 
 group of degree 1 2. The order of a sub-group of the group of degree 
 1 2 which keeps one symbol fixed is 2' . 31 This sub-group can have 
 no substitution consisting of a single cycle of 3 symbols, since no 
 such substitution can occur in the original group. Hence it must 
 permute the 11 symbols in two transitive sets of 9 and 2 symbols 
 respectively. It must therefore contain a self-conjugate sub-group 
 of order 2^ . 3' which keeps 3 of the 12 symbols unchanged; and this 
 sub-group must occur self-conjugately in 3 of the 12 sub-groups which 
 keep one symbol unchanged. This however makes the group of 
 degree 12 imprimitive, contrary to supposition. Hence this case 
 cannot occur. 
 
 Finally, then, the only possible values of x and y are a: = 1, 
 y = 0. The order of a group of degree 13, which has more than 
 one sub-group of order 13 and no odd substitutions, is 2*.3M3, 
 2* . 3^ 13, or 2^ 3^ 13. The order of a group of order 13 with odd 
 substitutions will be twice one of the preceding three numbers. 
 
 A further and much more detailed examination would be neces- 
 sary to determine whether groups of degree 13 correspond to any or 
 all of these orders. We shall see in Chapter XIV that there is a 
 group of degree 13 and order 2^ 3M3 ; and M. Jordan ^ states 
 that this is the only group of degree 13 which contains more than 
 one sub-group of order 13. 
 
 Ex. If n (> 3) and 2n + 1 are primes, shew that there is no 
 triply transitive group of degree 2n + 3 which does not contain the 
 alternating group. 
 
 146. As a further illustration, and for the actual value of 
 the results themselves, we proceed to determine all types of 
 primitive groups for degrees not exceeding 8. 
 
 1 Comptes Rendus, t. lxxv (1872), p. 1757. 
 
146] OF DEGREES THREE FOUR AND FIVE 205 
 
 (i) n = 3. 
 
 The symmetric group of 3 symbols has a single sub-group, 
 viz. the alternating group. Both these groups are necessarily 
 primitive. 
 
 (ii) n = 4. 
 
 Since a group whose order is the power of a prime cannot 
 (§ 124) be represented in primitive form, a primitive group of 
 degree 4 must contain 3 as a factor of its order. Hence the 
 only primitive groups of degree 4 are the symmetric and the 
 alternating groups. 
 
 (iii) 71 = 5. 
 
 Since 5 is a prime, every group of degree 5 is a primitive 
 group. The symmetric group of degree 5 contains 6 cyclical 
 sub-groups of order 5 ; therefore, by Sylow's theorem, every 
 croup of degree 5 must contain either 1 or 6 sub-groups of 
 order 5. Since the alternating group is simple, every sub- 
 group that contains 6 sub-groups of order 5 must contain the 
 alternating group. Hence, besides the alternating and the 
 symmetric groups, we have only sub-groups which contain a 
 sub-group of order 5 self-conjugately. In such a group, an 
 operation of order 5 can be permutable with its own powers 
 only. Hence (§112) the only sub-groups of the type in question 
 other than cyclical sub-groups, are groups of orders 20 and 10. 
 These are defined by 
 
 {(12345), (2354)}, 
 
 and {(12345), (25) (34)}. 
 
 (iv) n = 6. 
 
 If the order of a primitive group of degree 6 is not divisible 
 by 5, the order must (§ 141) be equal to or be a factor of 2^ . 3. 
 The order of a sub-group that keeps one symbol fixed is equal to 
 or is a factor of 2^ Hence the sub-group must keep two symbols 
 fixed, and therefore (§128) the group cannot be primitive. Hence 
 the order of every primitive group of degree 6 is divisible by 
 5, and every such group is at least doubly transitive. The 
 symmetric group contains 36 sub-groups of order 5 ; and hence, 
 since no transitive group of degree 6 can contain a self- 
 conjugate sub-group of order 5, every primitive group of degree 
 
206 PRIMITIVE GROUPS [146 
 
 6, M'hich does not contain the alternating group, must have 6 
 sub-groups of order 5. 
 
 If is such a group, the sub-group of Q that keeps one 
 symbol fixed is a transitive group of degree 5 which has a 
 self-conjugate sub-group of order 5. If this transitive group 
 of degree 5 were cyclical, every operation of the doubly transi- 
 tive group G of order 30 would displace all or all but one of the 
 symbols. Since 6 is not the power of a prime, this is impossible 
 (§ 105). Hence the sub-group of Q which keeps one symbol 
 fixed must be of one of the two types given above ; and the order 
 of G must be 120 or 60. Now we have seen, in § 142, that the 
 symmetric group of degree 6 has a single conjugate set of primi- 
 tive sub-groups of order 120 and a single set of order 60. Hence 
 there is a single type of primitive group of degree 6, correspond- 
 ing to each of the orders 120 and 60. These are defined by 
 
 {(126) (354), (12345), (2354)}, 
 and {(126) (354), (12346), (25) (34)}: 
 
 where the last two substitutions in each case generate a sub- 
 group that keeps one symbol unchanged. 
 
 (v) TO = 7. 
 
 Every transitive group of degree 7 is primitive ; and if it 
 does not contain the alternating group, its order must (§ 141) be 
 equal to or be a factor of 7 . 6 . 5 . 4. A cyclical sub-group of order 
 7 must (footnote, p. 201), in a group of degree 7 that contains 
 more than one such sub-group, be self-conjugate in a group of 
 order 21 or 42. Now neither 20 nor 40 is congruent to unity, 
 mod. 7 ; and therefore 5 cannot be a factor of the order of such 
 a group. Hence the order of a transitive group of degree 7, 
 that does not contain the alternating group, is equal to or is a 
 factor of 7.6.4. But 8 is the only factor of 7.6.4 which is 
 congruent to unity, mod. 7 ; and therefore, if the group contains 
 more than one sub-group of order 7, its order must be equal to 
 7.6.4 and it must contain 8 sub-groups of order 7. 
 
 Such a group must be doubly transitive ; for if a sub-group 
 of order 24 interchanges the symbols in two intransitive 
 systems, it is easily shewn that the group would contain 
 substitutions displacing three symbols only, and therefore that 
 it would contain the alternating group. A sub-group of degree 
 
146] OF DEGREES SIX AND SEVEN 207 
 
 24, transitive in 6 sjonbols, must contain 4 sub-groups of order 
 3. For if it contained only one, it would necessarily have 
 circular substitutions of 6 symbols, and the group of order 
 7.6.4 would have a self-conjugate sub-group of order 7.6.2; 
 which is not the case. Hence the sub-groups of order 24 must 
 be simply isomorphic with the symmetric group of 4 symbols. 
 The actual construction of the group is now reduced to 
 a limited number of trials. A group of degree 6, simply 
 isomorphic with the symmetric group of 4 symbols, and con- 
 taining no odd substitutions, may always be represented in the 
 form 
 
 {(234) (567), (2763) (45)}; 
 
 and we have to find a circular substitution of the seven symbols 
 1, 2, 3, 4, 5, 6, 7 such that the group generated by it shall be 
 permutable with this group. Moreover since, in the required 
 group, every operation of order 3 transforms some operation of 
 order 7 into its square, we may assume without loss of generality 
 that the circular substitution of order 7 contains the sequence 
 ..12.. and is transformed into its own square by (234) (567). 
 There are only three circular substitutions satisfying these 
 conditions, viz. 
 
 (1235476), 
 
 (1236457), 
 and (1237465). 
 
 It appears on trial that the group generated by the first of 
 these is not permutable with the sub-group of order 24, while 
 the groups generated by the other two are. There are therefore 
 just two groups of order 7.6.4 which contain the given group 
 of order 24. Now in the symmetric group of 7 symbols, a 
 sub-group of order 7.6.4 must, from the foregoing discussion, 
 be one of a set of 30 conjugate sub-groups. These all ehter in 
 the alternating group ; and therefore, in that group, they must 
 form two sets of 15 conjugate sub-groups each. Each of these 
 contains 7 sub-groups of the type 
 
 {(234) (567), (2763) (45)}; 
 and the alternating group contains a conjugate set of 105 such 
 such sub-groups. Hence each sub-group of this set will enter 
 in two, and only in two, sub-groups of the alternating group of 
 
208 PRIMITIVE GROUPS [146 
 
 order 7.6.4; and in the symmetric group these two sub-groups 
 are conjugate. Finally then, the sub-groups of order 7.6.4 
 form a single conjugate set in the symmetric group. They are 
 defined by 
 
 {(1236457), (234) (567), (2763) (45)}, 
 
 the two latter substitutions giving a sub-group that keeps 
 one symbol fixed. 
 
 These groups are simple; for since they are expressed as 
 transitive groups of degree 7, there can be no self-conjugate 
 sub-group whose order divides 24, while it is evident that a 
 self-conjugate sub-group that contains an operation of order 7 
 must coincide with the group itself Also since there are 8 
 sub-groups of order 7, these groups can be expressed as doubly 
 transitive groups of eight symbols. 
 
 A group of degree 7, which has only one sub-group of order 
 7, must either be cyclical or be contained in the group of order 
 7 . 6 given by § 112. Such groups are defined by 
 
 {(1234567), (243756)}, 
 
 or {(1234567), (235) (476)}, 
 
 or {(1234567), (27) (45) (36)}. 
 
 The simple group of order 168, which here occurs as a 
 transitive group of degree 7, is the only simple group of that 
 order. For, if possible, let there be a simple group G of order 
 168 and of a distinct type from the above. It certainly cannot 
 be expressed as a group of degree 7 ; and therefore it must 
 have 21 sub-groups of order 8. If two of these sub-groups 
 have a common sub-group of order 4, it must be contained 
 self-conjugately (§ 80) in a sub-group of order 24 or 56; and 
 this is inconsistent with the suppositions made. If on the 
 other hand, 2 is the order of the greatest sub-group common 
 to two sub-groups of order 8, such a common sub-group of order 
 2 must, on the suppositions made, be self-conjugate in a sub- 
 group of order 12. But a group of order 12, which has a self- 
 conjugate operation of order 2, must have a self-conjugate 
 sub-group of order 3 ; and therefore G would only contain 7 
 sub-groups of order 3, and could be expressed as a group of 
 degree 7 ; contrary to supposition. No other supposition is 
 possible with regard to the sub-groups of order 8, since 21 is 
 
146] OF DEGREES SEVEN AND EIGHT 209 
 
 not congruent to unity, mod. 8. Hence, finally, there is no 
 simple group of order 168 distinct from the group of degree 7. 
 
 (vi) n = 8. 
 
 The order of a primitive group of degree 8, which does not 
 contain the alternating group, cannot (§ 141) be divisible by 5. 
 Suppose, if possible, that the order of such a group is 2"+' . 3 
 (a = 0, 1, 2, 3). A substitution of order 3 must consist of two 
 cycles; and therefore the sub-group of order 2''3, which keeps 
 one symbol fixed, must interchange the others in two intransi- 
 tive systems of 3 and 4 respectively. In this sub-group, a 
 sub-group of order 3 must be one of four conjugate sub-groups, 
 and therefore a is either 2 or 3. Now a group of order 2^ . 3 
 or 2^ . 3 is soluble, as is seen at once by considering the sub- 
 groups of order 2' or 2". Hence a primitive group of order 2'^ . 3 
 or 2^.3 must contain a transitive self- conjugate sub-group of 
 order 8, whose operations are all of order 2. 
 
 If 7 is a factor of the order of the group, the group must be 
 doubly transitive ; aud from the case of w = 7, it follows that 
 the possible orders are 8.7, 8.7.2, 8.7.3, 8.7.6, and 
 8.7.6.4. Moreover, for the orders 8.7.2 and 8.7.6, the 
 group contains odd substitutions and therefore it contains self- 
 conjugate sub-groups of order 8 . 7 and 8.7.3 respectively. 
 
 A simple group of order 8.7.3 is necessarily identical in 
 type with the group of this order determined on p. 208 ; 
 and a group of order 8.7.3, which is not simple, is certainly 
 soluble. Hence a composite group of order 8.7.3, and 
 a group of order 8.7.6 which does not contain a simple 
 sub-group of order 8.7.3, must both, if expressible as pri- 
 mitive groups of degree 8, contain transitive self-conjugate 
 sub-groups of order 8 whose operations are all of order 2. 
 With the possible exception then of groups of order 8.7.6.4, 
 the only primitive groups of degree 8, which do not contain a 
 self-conjugate sub-group of order 8, are the simple group of 
 order 8.7.3 and any group of 8.7.6 which contains this self- 
 conjugately. We have seen that the simple group of order 
 8.7.3 contains a single set of 8 conjugate sub-groups of order 
 21, and therefore it can be expressed in one form only as a 
 group of degree 8. A group of degree 8 and order 8.7.6, 
 B. 14 
 
210 PRIMITIVE GROUPS [146 
 
 which contains this self-conjugately, can occur only in one 
 form, if at all ; for, if it exists, it must be triply transitive, and 
 it must be given by combining the simple group with any 
 operation of order 2 which transforms one of its operations of 
 order 7 into its own inverse. That such a group does exist has 
 been shewn in § 113. These two groups are actually given by 
 
 {(15642378), (1234567), (243756)}, 
 
 and {(1627) (5438), (1234567), (235) (476)}; 
 
 where in each case the last two substitutions give a sub-group 
 that keeps one symbol fixed. 
 
 To avoid unnecessary prolixity we shall, in dealing with the 
 primitive groups of degree 8, which contain a transitive self- 
 conjugate sub-group of order 8 whose operations are all of 
 order 2, anticipate some of the results of the next Chapter. It 
 will there be shewn that, if is an Abelian group of order N, 
 and L a group of isomorphisms (§156) of G, a group of degree N 
 may be formed which has a transitive self-conjugate sub-group 
 simply isomorphic with G, while at the same time the sub- 
 group that keeps one symbol fixed is isomorphic with L (§ 158). 
 The group of isomorphisms of a group of order 8, whose 
 operations are all of order 2, will be shewn in Chapter XIV to 
 be identical with the simple group of order 168. This group 
 has a single set of conjugate sub-groups of each of the orders 7 
 and 21, but no sub-group of order 14 or 42. When expressed 
 as a group of degree 7, it has a single set of conjugate sub- 
 groups of order 12 (or 24) which leave no symbols unchanged. 
 There are therefore primitive groups of degree 8 containiug 
 transitive self-conjugate sub-groups of order 8 corresponding 
 to each of the orders 8.7, 8.7.3, 2^3, 2«.3, and 8.7.6.4; 
 and in each case there is a single type of such group. 
 
 It remains to determine whether there can be any type 
 of group, of degree 8 and order 8.7.6.4, other than that just 
 obtained. Such a group must be one of 15 conjugate sub- 
 groups in the alternating group of degree 8, and can therefore 
 itself be expressed as a group of degree 14. Since it 
 certainly cannot be expressed as a group of degree 7, the 
 group of degree 14 must be transitive. The order of the sub- 
 group, in this form, that keeps one symbol fixed is 2=.3. If 
 
146] OF DEGREE EIGHT 211 
 
 this keeps only one symbol unchanged, it must interchange the 
 remaining symbols in four intransitive systems of 3, 3, 3 and 4 
 respectively, since a substitution of order 3 must clearly consist 
 of 4 cycles. A group of order 2"* . 3 cannot however be so 
 expressed ; and therefore the sub-group that keeps one symbol 
 fixed must keep two fixed. The group of degree 14 is therefore 
 imprimitive, and the group must contain a sub-group of order 
 2* . 3. Moreover, since the group cannot be expressed as a 
 transitive group of degree 7, this sub-group of order 2" . 3 must 
 contain (§ 123) a sub-group which is self-conjugate in the group 
 itself. The order of this sub-group must be a power of 2 ; 
 since the group is primitive, it cannot be less than 2^ On the 
 other hand, the order cannot be greater than 2^ since the group 
 contains a simple sub-group of order 7.6.4. Hence finally, 
 there is no type of primitive group of degree 8 and order 
 8.7.6.4 other than that already obtained. 
 
 There is no difficulty now in actually constructing the 
 primitive groups of degree 8 which have a self-conjugate sub- 
 group of order 8. They are all contained in the group of order 
 8.7.6.4; and this may be constructed from a sub-group 
 keeping one symbol fixed, which is given on p. 208, by the 
 method of § 109. It will thus be found that the group in 
 question is given by 
 
 {(81) (26) (37) (45), (1236457), (234) (567), (2763) (45)}; 
 
 while the groups of orders 8.7.3 and 8 . 7 are given by omitting 
 respectively the last and the two last of the four generating 
 operations. 
 
 The construction of the two remaining groups, of order 2^ . 3 
 and 2" . 3, is left as an exercise for the reader. 
 
 It may be noticed that it has been shewn incidentally, in 
 discussing above the possibility of a second type of group of 
 degree 8 and order 8.7.6.4, that the alternating group of 
 degree 8 can be expressed as a doubly transitive group of 
 degree 15. 
 
 It may similarly be shewn that the alternating group of 
 degree 7 can be expressed as a doubly transitive group of 
 degree 15, and the alternating group of degree 6 as a simply 
 transitive and primitive group of degree 15. 
 
 14—2 
 
212 PROPERTIES OF [147 
 
 147. We have seen in § 105 that a doubly transitive group, 
 of degree n and order n{n — 1), can exist only when n is the 
 power of a prime. For such a group, the identical operation is 
 the only one which keeps more than one symbol unchanged. 
 We shall now go on to consider the sub-groups of a doubly 
 transitive group, of degree n and order n{n — \) m, which keep 
 two symbols fixed. The order of any such sub-group is m; 
 since the group contains operations changing any two symbols 
 into any other two, the sub-groups which keep two symbols 
 fixed must form a single conjugate set. 
 
 Suppose first that the sub-group, which keeps two symbols 
 unchanged, displaces all the other symbols. The sub-group that 
 keeps a and h unchanged cannot then be identical with that 
 which keeps c and d unchanged, unless the symbols c and d are 
 the same pair as a and h. Since there are \n{n — l) pairs 
 of n symbols, the conjugate set contains ^n{n — 1) sub-groups; 
 and each sub-group of order m keeping two symbols fixed 
 must be self-conjugate in a sub-group of order 2m, which 
 consists of the operations of the sub-group of order m and of 
 those operations interchanging the two symbols that the sub- 
 group of order m keeps fixed. 
 
 Suppose next that all the operations of a sub-group H, 
 which keeps two symbols fixed, keep x symbols fixed, while 
 none of the remaining n — x symbols are unchanged by all the 
 operations of H. From x symbols \x{x — 1) pairs can be 
 formed, and therefore the sub-group that keeps one pair un- 
 changed must keep ^x(x — V) pairs unchanged. In this case, 
 
 the conjugate set contains —^ =^ distinct sub-groups of order 
 
 m, and H is therefore self-conjugate in a group K of order 
 x(x—l) m. The operations of this sub-group which do not 
 belong to U interchange among themselves the x symbols that 
 are left unchanged by H. Now since the group itself is doubly 
 transitive, there must be operations which change any two of 
 these X symbols into any other two; and any such operation 
 being permutable with H must belong to K. Hence if we 
 consider the effect of K on the x symbols only which are left 
 unchanged by H, K reduces to a doubly transitive group of 
 
148] DOUBLY TRANSITIVE GROUPS 213 
 
 degree x and order x(x — 1). It follows that x must be a prime 
 or the power of a prime. 
 
 148. The preceding paragraph suggests the combinatorial 
 
 111 ( 71/ 1 I 
 
 problem of forming from n distinct symbols —y =-^ sets of x 
 
 OC ipC — X ) 
 
 symbols, such that every pair of symbols occurs in one set of x and 
 
 no pair occurs in more than one. 
 
 There is one class of cases in which a solution of this problem is 
 
 given immediately by the theory of Abelian groups. Let G be an 
 
 Abelian group of order ^™ where ^ is a prime, and type (1, 1, 
 
 »™ — 1 
 to m units). We have seen, in § 49, that G has =- sub-groups of 
 
 «"•_!. pm-l _ I 
 
 order p, and f^a — 1~ sub-groups of order jo^. Now any pair 
 
 of sub-groups of order p generates a sub-group of order p'', and there- 
 fore every pair of sub-groups of order p occurs in one and only one 
 sub-group of order p^. Moreover, every sub-group of order p'' contains 
 p + 1 sub-groups of order p. When ]} is a prime and m any in- 
 
 n™ — 1 
 teger, it is therefore always possible to form from =- symbols 
 
 p™ _ 1 . p™-i _ 1 
 
 ^ — - — sets of _p -f 1 symbols each, such that every pair of 
 
 the symbols occurs in one set of ^ 4- 1 and no pair occurs in more 
 than one set. 
 
 Supposing that, for given values of n and x, such a distribution 
 is possible, it is still of course an open question as to whether there 
 is a doubly transitive substitution group of the w symbols, such that 
 every substitution which keeps any two symbols unchanged keeps 
 also unchanged the whole set of x in which they occur. When x is 
 greater than 3, the question as to the existence of such groups is 
 one which still remains to be investigated. There is however an 
 important class of groups, to be considered later (Chapter XIV), 
 that possess a closely analogous property. These groups are doubly 
 transitive ; and from the n symbols upon which they operate, we can 
 
 form — =^ sets of x, that are interchanged transitively by the 
 
 substitutions of the group: the sets being such that every pair 
 occurs in one set and no pair in more than one set. 
 
 If n{n-'[)m is the order of such a group, and ii H ia a, 
 sub-group of order m which keeps a given pair fixed, then H must 
 interchange among themselves the remaining as - 2 symbols of that 
 set of X which contains the pair kept unchanged by H. H contains, 
 as a self -conjugate sub-group, the group h which leaves every symbol 
 of the set of x unchanged ; and if m" is the order of this sub-group, 
 while m = mlm!', then ?»' is the order of the group to which H 
 
214 PROPERTIES OF [148 
 
 reduces when we consider its effect only on the x-2 symbols. Now 
 h is self-conjugate in the group K that interchanges all the symbols 
 of the set of x among themselves. But since the original group is 
 doubly transitive, it must contain substitutions which change any 
 two of the set of x into any other two, and every such substitution 
 must belong to K Hence K must be doubly transitive in the x 
 symbols, and therefore finally the order of the group, to which K 
 reduces when we consider its effect on the x symbols only, is 
 x{x-l)m'. Since the order of h, which keeps unchanged each of 
 the X symbols is m", the order oi E ia x(x-l)m. 
 
 149. When x = 3, n must be of the form 6k + I or 6k + 3, since 
 
 otherwise — — - would not be an integer. The substitutions of 
 
 a doubly transitive group of degree n, which possesses a complete 
 set of -J- w (m - 1) triplets, must be such that every substitution which 
 leaves two given symbols unchanged also leaves a third definite 
 symbol unchanged. 
 
 The smallest possible value of ?i is 7 ; and the group of Ex. 2, 
 § 35, which is one of the groups obtained in § 146, satisfies all the 
 conditions. In fact, the group is clearly a doubly transitive group 
 of degree 7 ; and since its order is 7.6.4, the order of the sub- 
 group which keeps two symbols fixed is 4. Now in the sub-group 
 
 {(267) (345), (23) (47)}, 
 which keeps one symbol fixed, the only operations that keep 1 and 
 2 unchanged are 
 
 (35) (67), (36) (57), (37) (56). 
 
 These with identity form a sub-group of order 4, which must 
 therefore be the sub-group that keeps three symbols fixed. The 
 complete set of triplets in this case is 
 
 124, 137, 156, 235, 267, 346, 457. 
 
 The next smallest value of n is 9, and in this case again, a group 
 with the required properties exists. 
 
 Ex. Shew that the group 
 
 {(26973854), (456) (798)} 
 is an imprimitive group of order 48, each imprimitive system 
 containing two symbols ; and that the sub-group, which keeps the 
 symbols of one imprimitive system unchanged, is isomorphic with 
 the symmetric group of three symbols. Prove that this group is 
 permutable with 
 
 {(123) (456) (789), (147) (258) (369)}, 
 and thence that 
 
 {(123) (456) (789), (26973854), (456) (798)}, 
 is a doubly transitive group of degree 9, which possesses a complete 
 set of 1 2 triplets. 
 
150] DOUBLY TRANSITIVE GKOUPS 215 
 
 The reader is not to infer from the examples given that, when 
 n is of the form 6^ + 1 or 6A + 3, there is always a doubly transitive 
 group of degree n which possesses a complete set of triplets. It is 
 a good exercise to verify that there is no such group when n is 13. 
 
 The case n = 13, x= 4: is the simplest case that can occur of the 
 division of n symbols into sets of x in the manner of § 148 when x 
 is greater than 3. We shall see in Chapter XIV that there is a 
 doubly transitive group of degree 13 such that from the 13 symbols 
 permuted by the group a complete set of 13 quartets can be 
 formed, which are themselves permuted by the operations of the 
 group. Of the operations forming a sub-group that keeps two given 
 symbols fixed, half will keep fixed the two other symbols, which form 
 a quartet with the two given symbols, and half will permute them. 
 
 On the question of the independent formation of a complete set 
 of triplets of n symbols, and in certain cases of the group of 
 degree n which interchanges the triplets among themselves, refer- 
 ence may be made to the memoirs mentioned in the subjoined 
 footnote". 
 
 150. We shall conclude the present Chapter with some 
 applications of substitution groups, which enable us to complete 
 and extend certain earlier results. 
 
 We have seen in § 107 that the substitutions of n symbols, 
 which are permutable with each of the substitutions of a 
 regular substitution group of order n of the same n symbols, 
 form another regular substitution group of order n ; and that, 
 if G is Abelian, the latter group coincides with G. Hence the 
 only substitutions of n symbols, which are permutable with a 
 circular substitution of the n symbols, are the powers of the 
 circular substitution. 
 
 Let now *S be a regular substitution of order m, in mn 
 symbols. It must permute the symbols in n cycles of m 
 symbols each; and so we may take 
 
 If T is permutable with S, and if it changes a,.^ into arg, it 
 clearly must permute the m symbols 
 
 CCfi , Clj-2 , , Clrm 
 
 1 Netto: " Substitutionentheorie," pp. 220—235; " Zur Theorie der Tripel- 
 systeme," Math. Ann. Vol. xlii, (1892), pp. 143 — 152. Moore: " Couoeruing 
 triple systems," i¥aS7i. Ann. Vol. xliii, (1893), pp. 271—285. Heffter : "Ueber 
 Tripelsysteme," Math. Aim., Vol. xlix, (1897), pp. 101-^112. 
 
216 SUBSTITUTIONS PERMUTABLE WITH [150 
 
 among themselves ; and therefore, so far as regards its effect on 
 these m symbols, T must be a power of 
 
 Again, if T changes arp into agq, it must change the set 
 
 (In , dn , , drm > 
 
 into the set 
 
 as otherwise it would not be permutable with S. 
 
 Now the totality of the substitutions of the mn symbols, 
 which are permutable with 8, form a group G^- This group 
 must, from the properties of T just stated, be imprimitive, 
 interchanging the symbols in n imprimitive systems of m 
 symbols each ; and the symbols in any cycle of 8 will form an 
 imprimitive system. Moreover, the self-conjugate sub-group 
 Hg of this group, which permutes the symbols of each system 
 among themselves, is the group of order m" generated by 
 
 \fli\(hi (h.m)j (Cl2i'^22 (^wn)t > (P'm'^mi Ojnm)- 
 
 In fact, every substitution of this group is clearly per- 
 mutable with ;S; and conversely, every substitution of the mn 
 symbols, which does not permute the systems, must belong to 
 this group. 
 
 C 
 Now jY' is capable of representation as a group of degree 
 
 n, for none of its operations changes every one of the n systems 
 
 C 
 into itself. Hence n\ is the greatest possible order of „^. On 
 
 ■"s 
 the other hand, every operation of the group, generated by 
 
 (O^liCtai (^ni) \^2^22 ^nV \P^ini^2^n ^nm/ 
 
 and (an a^) (oia a^^ (oim a^i), 
 
 is clearly permutable with 8; and this group, being simply 
 isomorphic with the group 
 
 [{0,-^0,2 an), {(ha.^)}, 
 
 i.e. with the symmetric group of n symbols, is of order n !. 
 
 Hence, finally, the order of Os is m".w!; and Gs is gene- 
 rated by 
 
 (ffliifflji (Itni) (0^12^22 ^m) (flimO-om (^jim/i 
 
 (flnd^i) (.(^12 ^22^ (t^imauin,), 
 
 and (<IiiKi2 ttim). 
 
151] EVERY SUBSTITUTION OF A GEOUP 217 
 
 151. Let hr be a regular substitution group of order m in 
 the m symbols 
 
 and let Sn be one of its substitutions. Then if for r we write 
 
 in turn 1, 2, ,n, and if for each value of t from 1 to m we 
 
 form the substitution 
 
 SitS^t Snt, 
 
 the set of m substitutions so formed constitute an intransitive 
 group H in the mn symbols, simply isomorphic with h^. 
 
 The method of § 150 can be applied directly to determine 
 the group Og of degree mn, each of whose substitutions are 
 permutable with every substitution of H. The order of this 
 group is m".m!; and it can be generated by 
 
 yCtiiCt^ Ctni) V^12'^22 ^jiaj \(^i7n^2m ^n7n)> 
 
 \Ci11Ci21) v^2^22.) V'^im.^2m/j 
 
 and W ; 
 
 where V is the regular group in the symbols 
 
 Ct^i, 0,12, ) (f'lmi 
 
 each of whose substitutions is permutable with every substitu- 
 tion of hi . 
 
 This group will contain H if, and only if, H is an Abelian 
 group. Moreover, the only self-conjugate substitutions of Gjj 
 are the substitutions of H contained in it. For if Gjj 
 contained other self-conjugate substitutions 8^, 82, ..., every 
 operation of Gg would be permutable with every operation of 
 the group {H, 81, 8^, ..:}. Now Gjj is transitive, so that 
 81, 82, ... must displace all the symbols; and therefore 
 [B, Si, 82, ...} has all its substitutions regular in the mn 
 symbols. If its order is m«i, where n = WiWj, the order of the 
 group formed of all the substitutions of mn symbols, which are 
 permutable with each of its operations, is (mn^^^ . Wa ! ; and this 
 number is less than m" .n\. Thus the supposition, that Gj^ has 
 self-conjugate operations other than the operations of H which 
 it contains, leads to an impossibility. 
 
 By means of this and the preceding section, the reader will 
 have no difficulty in forming the group of n symbols, which is 
 permutable with every operation of any given group in the n 
 symbols. 
 
218 SUBSTITUTION GROUPS WHOSE ORDERS [152 
 
 152. If a group, whose order is a power of a prime f, be 
 expressed as a transitive substitution group, its degree must 
 also be a power of p (| 123). Moreover such a group, since it 
 has self-conjugate operations, must necessarily be imprimitive. 
 
 The greatest value of m, for which a group of order p^ can 
 be expressed as a transitive group of degree p", where n is 
 regarded as given, is determined at once by considering the 
 symmetric group of degree p^. The highest power of p that 
 divides p"! is p", where 
 
 ^ ^ pn-ij^ pn-2 j^ +^ + 1. 
 
 Hence the symmetric group of degree p" contains a set of 
 conjugate sub-groups of order p" and it contains no groups whose 
 order is a higher power of p. Also, these groups are transitive in 
 the p" symbols ; for any one of them must contain a circular 
 substitution of order p". There are therefore groups of order p" 
 which can be expressed as transitive groups of degree p^ ; but 
 no group of order p"' {v > v) can be so expressed. Moreover, in 
 order that a group of order p™ may be capable of representation 
 as a transitive group of degree p", it must be simply isomorphic 
 with a transitive sub-group of the above substitution group of 
 order p". 
 
 This group may be constructed synthetically as follows. 
 Since the sub-group of order p"^", that leaves one symbol 
 unchanged, is contained in a sub-group of order p''-^+'^ (| 55), 
 there must be imprimitive systems containing p symbols each. 
 If then we distribute the _p" symbols into p^~^ sets of p each, 
 and with each set of p form a circular substitution, the p""' 
 permutable and independent circular substitutions will generate 
 an intransitive group of order pp" . It will be the self-conju- 
 gate sub-group of the group of order p", which permutes the 
 symbols of each system among themselves. 
 
 Again, the s^^stems of p symbols each may be taken in sets 
 of p to form systems of p'' symbols each, since the previously 
 considered sub-group of order ^"-"+1 must be contained in a sub- 
 group of order ^"-"+2. Hence w e may form syste ms of p^ symbols 
 by combining the previous systems in sets of p ; and then with 
 each p^ symbols we can form a circular substitution, whose p"" 
 power is the product of the p circular substitutions of order p, 
 
153] ARE POWERS OF PRIMES 219 
 
 which have been previously formed from the p'^ symbols. The 
 symbols of any set of ^^ will then be interchanged by a transitive 
 group of order p^+^ ; and since there are p^~^' such sets, we obtain 
 in this way an intransitive group of ^p""'-h2'""". The group thus 
 formed is that self-conjugate sub-group of the original group, 
 which interchanges among themselves the symbols of each 
 system of p'. This process may be continued, taking greater 
 and greater systems, till at the last step we combine the p 
 systems of p"~^ symbols each into a single system by means of 
 a circular substitution of order p". The order of the resulting 
 group is clearly p", as it should be. 
 
 The self-conjugate operations of this group form a sub-group 
 of order p. 
 
 For suppose, if possible, they form a sub-group of order p'^. 
 Every operation of this sub-group displaces all the symbols ; 
 and therefore, when expressed as a substitution group in the p^ 
 symbols, it must interchange them transitively in p^"^ sets of 
 p"" each. 
 
 Now (§ 151) those substitutions of the p^ symbols, which 
 are permutable with every operation of this sub-group, form a 
 group of order ^'■^"~'.p"~^!; this number is only divisible by 
 p", as it must be, when r= 1. 
 
 Ex. Shew that, for the group of degree p" and order ^^+', the 
 
 TJ 
 
 factor groups -~^ (of § 53) are all of type (1) except the last, 
 which is of type (1, 1). 
 
 The fact that v is a function of p when n is given, explains 
 why, in classifying all groups of order p", some of the lower 
 primes may behave in an exceptional manner. Thus we saw, in 
 § 73, that for certain groups of order p'^ it was necessary to 
 consider separately the case _p = 3. The present article makes 
 it clear that, while there may be more than one type of group 
 of order p'^ (p > 3), which can be expressed as a transitive group 
 of degree p^, there is only a single type of group of order 3'' 
 which can be expressed transitively in 9 symbols. 
 
 153. In the memoirs referred to in the footnote on p. 155, 
 M. Mathieu has 'demonstrated the existence of a remarkable group, 
 
220 EXAMPLE 
 
 of degree 12 and order 12 . 11 . 10 . 9 . 8 . , which is quintuply trans- 
 itive. The generating operations of this group have been given in 
 the note to § 108, at the end of Chapter VIII. The verification of 
 some of the more important properties of this group, as stated in 
 the succeeding example, forms a good exercise on the results of this 
 and the two preceding Chapters. 
 
 Ex. Shew that the substitutions 
 
 (1254) (3867), (1758) (2643), 
 
 (12) (48) (57) (69), (a2) (58) (46) (79), 
 
 (a6) (57) (68) (49), (6c) (47) (58) (69), 
 
 generate a quintuply transitive group of degree 12 and order 
 
 12.11.10.9.8. 
 
 Prove that this group is simple ; that a sub-group of degree 1 1 
 and order 11.10.9.8, which leaves one symbol unchanged, is a 
 simple group ; and that a sub-group of degree 10 and order 10.9.8, 
 which leaves two symbols unchanged, contains a self-conjugate sub- 
 group simply isomorphic with the alternating group of degree 6. 
 
 Shew also that the group of degree 12 contains (i) 1728 
 sub-groups of order 11 each of which is self-conjugate in a group 
 of order 55 ; (ii) 2376 sub-groups of order 5, each of which is self- 
 conjugate in a group of order 40 : (iii) 880 sub-groups of order 27, 
 eacli of which is self-conjugate in a group of order 108 : (iv) 1485 
 sub-groups of order 64. 
 
 Prove further that the group is a maximum sub-group of the 
 alternating group of degree 12. 
 
CHAPTER XI. 
 
 ON THE ISOMORPHISM OF A GROUP WITH ITSELF. 
 
 154. It is shewn in § 24 that, if all the operations of a 
 group are transformed by one of themselves, which is not self- 
 conjugate, a correspondence is thereby established among the 
 operations of the group which exhibits the group as simply 
 isomorphic with itself 
 
 In an Abelian group every operation is self-conjugate, and 
 the only correspondence established in the manner indicated is 
 that in which every operation corresponds to itself If however 
 in an Abelian group we take, as the operation which corresponds 
 to any given operation S, its inverse 8~'^, then to ST will 
 correspond S~^T~^ or {ST)"'^; and the correspondence exhibits 
 the group as simply isomorphic with itself For this particular 
 correspondence, a group of order 2 is the only one in which 
 each operation corresponds to itself 
 
 It is therefore possible for every group, except a group of 
 order 2, to establish a correspondence between the operations of 
 the group, which shall exhibit the group as simply isomorphic 
 with itself Moreover, we shall see that in general there are 
 such correspondences which cannot be established by either of 
 the processes above given. We devote the present Chapter to 
 a discussion of the isomorphism of a group with itself It will 
 be seen that, for many problems of group-theory, and in par- 
 ticular for the determination of the various types of group 
 which are possible when the factor-groups of the composition- 
 series are given, this discussion is most important. 
 
222 GROUP OF ISOMORPHISMS [155 
 
 155. Definition. A correspondence between the opera- 
 tions of a group, such that to every operation S there 
 corresponds a single operation 8', while to the product 8T of 
 two operations there corresponds the product S'T' of the 
 corresponding operations, is said to define an isomorphism of 
 the group with itself. 
 
 That isomorphism in which each operation corresponds to 
 itself is called the identical isomorphism. In every isomorphism 
 of a group with itself, the identical operation corresponds to 
 itself; and the orders of two corresponding operations are 
 the same. For if 1 and S were corresponding operations, so 
 also would be 1.1 and 8'^; and therefore more than one operation 
 would correspond to 1. Again, if 8 and 8', of orders n and n', 
 are corresponding operations, so also are *S" and 8''" ; and there- 
 fore n must be a multiple of n'. Similarly n' must be a multiple 
 of n ; and therefore n and n' are equal. 
 
 If the operations of a group of order N are represented by 
 
 1, Oi, O2, , Ojf_i, 
 
 and if, for a given isomorphism of the group with itself, 8r is 
 
 the operation that corresponds to 8r (r = 1, 2, , N—1), the 
 
 isomorphism will be completely represented by the symbol 
 
 1, Oi, O2, , Ojv'-l 
 
 1, Oi , O2, , O if—i 
 
 In this symbol, two operations in the same vertical line are 
 corresponding operations. When no risk of confusion is 
 thereby introduced, the simpler symbol 
 
 18'^ 
 is used. 
 
 156. An isomorphism of a group with itself, thus defined, 
 is not an operation. The symbol of an isomorphism however 
 defines an operation. It may, in fact, be regarded as a substi- 
 tution performed upon the iV letters which represent the 
 operations of the group. Corresponding to every isomorphism 
 there is thus a definite operation ; and it is obvious that the 
 operations, which correspond to two distinct isomorphisms, are 
 
156] 
 
 OF A GIVEN GROUP 
 
 223 
 
 themselves distinct. The totality of these operations form a 
 group. For let 
 
 '8' 
 S' 
 
 and 
 
 'S'' 
 S" 
 
 be any two isomorphisms of the group with itself. Then if, as 
 hitherto, we use curved brackets to denote a substitution, we 
 have 
 
 [s'j [s") " [s"J ■ 
 
 But since* 
 
 -s~ 
 
 _8'_ 
 
 is 
 
 an isomorphism, the relation 
 
 
 
 8p8q = 8,. 
 
 requires that 
 
 
 8p 8q' = 8r'. 
 
 And since 
 
 -s'- 
 
 _8"_ 
 
 is 
 
 an isomorphism, the relation 
 
 
 
 8p 8q = *Si,.' 
 
 requires that 
 
 
 a f/ cf /f c! // 
 Wp Oq =i^r ■ 
 
 Hence if 
 
 
 8pSq = 8r, 
 
 then 
 
 
 Op Oq = Or , 
 
 and therefore 
 
 -8 ' 
 
 is 
 
 an isomorphism. 
 
 The product of the substitutions which correspond to two 
 isomorphisms is therefore the substitution which corresponds to 
 some other isomorphism. 
 
 The set of substitutions which correspond to the isomor- 
 phisms of a given group with itself, therefore form a group. 
 
 Definition. A group, which is simply isomorphic with 
 the group thus derived from a given group, is called the 
 group of isomorphisms of the given group. 
 
 It is not, of course, necessary always to regard this group as 
 a group of substitutions performed on the symbols of the 
 operations of the given group. But however the group of 
 isomorphisms may be represented, each one of its operations 
 corresponds to a definite isomorphism of the given group. To 
 avoid an unnecessarily cumbrous phrase, we may briefly apply 
 
224 COGEEDIENT AND CONTRAGREDIENT [156 
 
 the term " isomorphism " to the operations of the group of 
 isomorphisms. So long, at all events, as we are dealing with 
 the properties of a group of isomorphisms, no risk of confusion 
 is thereby introduced. Thus we shall use the phrase "the 
 
 isomorphism [„,]'' as equivalent to "the operation of the 
 
 group of isomorphisms which corresponds to the isomorphism 
 
 8' 
 
 157. If S is some operation of a group G, while for S each 
 operation of the group is put in turn, the symbol 
 
 defines an isomorphism of the group. For if 
 
 then t-'8p'^ . 2-^>S,S = 'Z-'8j,SgX = S-^/Sf,2 ; 
 
 and S~^/Sr2 is an operation of the group. An isomorphism of a 
 
 group, which is thus formed on transforming the operations of the 
 
 group by one of themselves, is called a cogredievt isomorphism. 
 
 / Sf \ 
 All others are called contragredient isomorphisms^ If „, j is 
 
 any contragredient isomorphism, the isomorphisms 
 
 when for S each operation of the group is taken successively, 
 are said to form a class of contragredient isomorphisms. 
 
 Theorem I. The totality of the cogredient isomorphisms of 
 a group Q form a group isomorphic with G; this group is 
 a self-conjugate sub-group of the group of isomorphisms of G^- 
 
 The product of the isomorphisms 
 
 1 Klein, " Vorlesungen iiber das Ikosaeder und die Auflosung der Gleioh- 
 ungen vom fiinften Grade" (1884), p. 232. Holder, Math. Ann., Vol. xliii 
 (1893), p. 314. 
 
 2 Holder, "Bildung zusammengesetzter Gruppen," Math. Ann., Vol. xlv: 
 (1895), p. 326. 
 
157] ISOMORPHISMS 225 
 
 is given by 
 
 where S2' = 2". 
 
 The product of two cogredient isomorphisms is therefore an- 
 other cogredient isomorphism; hence the cogredient isomorph- 
 isms form a group. Moreover, if we take the isomorphism 
 
 [-z-'sx) 
 
 as corresponding to the operation 2 of the group G, then to 
 every operation of G there will correspond a definite cogredient 
 isomorphism, so that to the product of any two operations 
 of G there corresponds the product of the two corresponding 
 isomorphisms. The group G and its group of cogredient 
 isomorphisms are therefore isomorphic. If G contains no 
 self-conjugate operation, identity excepted, no two isomorphisms 
 corresponding to different operations of G can be identical ; 
 and therefore, in this case, G is simply isomorphic with its 
 group of cogredient isomorphisms. If however G contains self- 
 conjugate operations, forming a self-conjugate sub-group H, 
 then to every operation of H there corresponds the identical 
 isomorphism ; and the group of cogredient isomorphisms is 
 
 G 
 simply isomorphic with ^. 
 
 Let now 
 be any isomorphism. Then 
 
 is'. 
 
 ^>S^-V 8 \(S\_(8'\( 8 \/8 
 
 \.%-'8%)\8')~\s)\t-^8tj{8'. 
 
 8' \ I 'Z-'8X 
 
 iir 
 
 = [■Z'-'8'Z') • 
 
 15 
 
t 
 
 226 THE HOLOMORPH [157 
 
 The isomorphism ( „, ) therefore transforms every cogredient 
 
 isomorphism into another cogredient isomorphism. It follows 
 that the group of cogredient isomorphisms is self-conjugate 
 within the group of isomorphisms. 
 
 158. Let Ghe a, group of order J\^, whose operations are 
 
 1, Oi, O2, , ^tr—i'-, 
 
 and let L be the group of isomorphisms of G. We have seen 
 in § 20 that G may be represented as a transitive group of 
 substitutions performed on the N symbols 
 
 1, Oi, O2, , Ojv— 1; 
 
 and that, when it is so represented, the substitution which 
 corresponds to the operation S^ is 
 
 1, Oi, O2, , 'JJV— 1 
 
 \Ox, l^i^x, O^^xj ) ^^N—i^x 
 
 or more shortly 
 
 ssx) ■ 
 
 When G is thus represented, we will denote it by G'. We 
 have already seen that L can be represented as an intransitive 
 substitution group of the same N symbols ; a typical substitu- 
 tion of L, when it is so represented, is 
 
 /I, Sj, Sr^, , /Sjsr_i\ 
 
 \1, Si, S^', , S'j^-J ' 
 
 or more shortly ( o' ) ■ 
 
 When L is thus represented, we will denote it by L'. It is 
 clear that the two substitution groups G' and L' have no 
 substitution in common except identity. For every substitution 
 of L' leaves the symbol 1 unchanged; and no substitution of 
 G', except identity, leaves 1 unchanged. 
 
 ^"" © ' iss) © = issx) {s'sl) 
 
 S' 
 S'Sx 
 
 ( ^\ 
 
 \SSx. 
 
158] OF A GROUP 227 
 
 Every operation of L' is therefore permutable with G'. 
 Hence if M is the order of L, the group {G-',L'\, which we will 
 call K', is a transitive group of degree N and order NM, 
 
 containing Q' self-conjugately. Further f „,] transforms [ „„ 
 
 / Si \ 
 
 into ( oa /] ; and these two substitutions of Q' correspond to 
 
 the operations 8^ and 8^ of Q. Hence the isomorphism, 
 established on transforming the substitutions of C by any 
 
 /Of N 
 
 substitution ( „,j of L' , is the isomorphism denoted by the 
 symbol L, 
 
 8 
 
 Since ( „ ao _. ) i^ ^ substitution of L' , the substitution 
 
 \a QQ -i) ( OQ ) ' °'' ( <? q) ' belongs to K'. Hence K' contains 
 the set of substitutions 
 
 ;sS)' (^=0'i'2, ,N-n 
 
 These form (§ 107) a transitive group G", simply iso- 
 morphic with G' and such that every substitution of G" is 
 permutable with every substitution of G'. Moreover (Z.c), 
 the substitutions of G" are the only substitutions of the N 
 symbols which are permutable with each of the substitutions 
 of(?'. 
 
 Suppose now that S is any substitution of the N symbols 
 which is permutable with G' . When the substitutions of 
 G' are transformed by 2, the resulting isomorphism is iden- 
 
 tical with that given by some substitution, say ( „, j , of L'. 
 
 Hence S f „, 1 is a substitution of the iV symbols which is 
 
 permutable with every substitution of G'. It therefore belongs 
 to G"; and hence 2 belongs to K'. It follows that K' contains 
 every substitution of the N symbols which is permutable 
 with G'. 
 
 15—2 
 
228 PROPERTIES OF [158 
 
 The only substitutions common to G' and Q" are the self- 
 
 K' 
 
 conjugate substitutions of either. The factor group 
 
 is simply isomorphic with — , where g is the group of cogredient 
 
 isomorphisms of contained in L. The groups Q' and 0" are 
 identical only when G' is Abelian ; in this case, g consists of the 
 identical operation alone. 
 
 Definition. A group K, simply isomorphic with the 
 substitution group K' which has just been constructed, we 
 shall call the holomorph of G. 
 
 159. An isomorphism must change any set of operations, 
 which are conjugate to each other, into another set which are 
 conjugate. For if 
 
 . © 
 
 be the isomorphism, and if 
 
 then s,'-'S^'s; = s;, 
 
 so that 8x' and Sy are conjugate operations when So; and Sy are 
 conjugate. A cogredient isomorphism changes every set of 
 conjugate operations into itself; and all the members of a class 
 of contragredient isomorphisms permute the conjugate sets in 
 the same way. If 
 
 is an isomorphism which changes every conjugate set of opera- 
 tions of G into itself, and if 
 
 ■8' 
 
 & 
 
 is any isomorphism of G, then the isomorphism 
 
 U'7 Uv \8") 
 changes every conjugate set into itself It follows that those 
 isomorphisms, which change every conjugate set of operations 
 into itself, form a self-Conjugate sub-group of the complete group 
 of isomorphisms. This sub-group clearly contains the group of 
 cogredient isomorphisms and may be identical with it. 
 
160] CONTRAGREDIENT ISOMORPHISMS 229 
 
 /On 
 
 If now f „, j is any isomorphism of G of order n, the 
 substitutions 
 
 © 
 
 ^andL^), (a;=0, 1 ,N-1), 
 
 generate a group of order Nn. When J is used to represent 
 the isomorphism, this group may be denoted by [J, G] ; as shewn 
 above, it contains G self-conjugately. Suppose that n is prime 
 and not a factor of N. The operation / is not permutable 
 with every operation of G ; and therefore there must be opera- 
 tions S oi G which are permutable with no operation of the 
 conjugate set to which J belongs. The number of operations 
 which in (/, G} are conjugate to such an operation S must be a 
 multiple of n ; and since n is not a factor of N, this conjugate 
 set of operations must be made up of n distinct conjugate sets 
 of operations in G. The isomorphism J must therefore inter- 
 change some of the conjugate sets of G. 
 
 The same result is clearly true if the order n oi J has any 
 prime factor not contained in N. Hence : — 
 
 Theorem II. An isomorphism of a group G, whose order 
 contains a prime factor which does not occur in the order of G, 
 must interchange some of the conjugate sets of G. 
 
 „,) or J" leaves no operation 
 
 except identity unchanged, it must in {/, G] be one of N 
 conjugate operations. For if 
 
 J would be permutable with Sy8x~^, which is not the case. 
 These N conjugate operations are 
 
 J, Jbi, t/02! ) t/Ojy_l, 
 
 and since the first transforms every operation of G, except 
 identity, into a different one, the same must be true of all the 
 set. If now J transformed any operation 8 into a conjugate 
 operation 1,-^8%, JX"^ would transform 8 into itself ; hence 
 J must transform every conjugate set of G into a different 
 conjugate set. 
 
230 THE GROUP OF ISOMORPHISMS [160 
 
 The special case in which the order of J is two may here be 
 considered. Representing the N operations conjugate to J by 
 
 J, Ji, J-i, J Js-\i 
 
 the N operations of G are 
 
 J , JJi, JJi, , JJ]g-i. 
 
 Now J'-'.JJ^.J=J^J={JJ^-\ 
 
 so that J transforms every operation of G into its own inverse 
 But if 
 
 S' = s-\ 
 
 and T' = T-\ 
 
 then S'T' = S-^T--' = {TS)~\ 
 
 Now as S'T' is the operation into which the isomorphism 
 transforms ST, it must be {ST)~'^, and therefore 
 
 ST= TS. 
 
 The group G is therefore an Abelian group of odd order. 
 
 161. Any sub-group H oi G is transformed by an isomorph- 
 ism into a simply isomorphic sub-group H' ; but H and H' 
 are not necessarily conjugate within G. If however the set of 
 conjugate sub-groups 
 
 are the only sub groups of (? of a given type, every isomorph- 
 ism must interchange them among themselves ; and if no 
 isomorphism transforms each one of the set into itself, the 
 group of isomorphisms can be represented as a transitive group 
 of degree m. 
 
 Suppose now that no operation of G is permutable with 
 each of the conjugate sub-groups 
 
 -"!) -"2) 3 -^mj 
 
 SO that G, or what in this case is the same thing (since G can 
 have no self-conjugate operation except identity) the group of 
 cogredient isomorphisms of G, can be expressed as the transitive 
 group of m symbols that arises on transforming the set of m 
 sub-groups by each operation of G. Let / be any operation, of 
 order fi, that transforms G and each of the set of m conjugate 
 sub-groups, into itself. Then J^ is the lowest power of / that 
 can occur in G, since no operation of G transforms each of the m 
 
161] REPRESENTED IN TRANSITIVE FORM 231 
 
 sub-groups into itself. Now in {J, 0}, the greatest sub-group 
 that contains H^ self-conjugately is [J, 1^}, Ir being the greatest 
 sub-group of that contains H^ self-conjugately. Also, in 
 (/, G} the set of sub-groups {J, I^}, (r=l, 2, ..., m), is a 
 complete conjugate set. Now the set of groups 
 
 have by supposition no common operation except identity ; and 
 therefore the greatest common sub-group of 
 
 {/,/,}, {/,/,}, , {J, 74 
 
 is {/}. Hence {J} is a self-conjugate sub-group of {J, G} ; and 
 since G is also a self-conjugate sub-group of {J, G], while {/} 
 and G have no common operation except identity, J must be 
 permutable with every operation of G. Every operation 
 therefore which is permutable with G, and with each of the 
 sub-groups 
 
 -"Ij -"2) > ^m> 
 
 is permutable with every operation of G. Thus finally, no 
 contragredient isomorphism can transform each of the sub- 
 groups Rr(r =1,2, , m) into itself Hence: — 
 
 Theorem III. If the conjugate set of m sub-groups 
 
 -"l) -"2; ) -"m 
 
 contains all the sub-groups of G of a given type, and if no 
 operation of G is permutable with each sub-group of the set, the 
 group of isomorphisms of G can be represented as a transitive 
 group of degree m. 
 
 Corollary I. If G contains kp + 1 sub-groups of order p", 
 where p'^ is the highest power of a prime p that divides the 
 order of G; and if the greatest sub-group I, that contains a 
 sub-group of order p"^ self-conjugately, contains no self-conju- 
 gate sub-group of G ; then the group of isomorphisms of G can 
 be represented as a transitive group of degree kp-^1. 
 
 For it has been seen that the groups of order p" contained 
 in G form a single conjugate set. 
 
 Corollary II. If the conjugate set of m sub-groups of G 
 is changed into itself by every isomorphism of G, and if no 
 
232 CHARACTERISTIC SUB-GROUPS [161 
 
 operation of G is per mutable with every one of these sub- 
 groups ; then the group of isomorphisms of G can be expressed 
 as a transitive group of degree m. 
 
 In fact, under the conditions stated, the reasoning applied 
 to prove the theorem may be used to shew that no isomorphism 
 of G can transform each of the m sub-groups into itself 
 
 162. Definition. Any sub-group of a group G which is 
 transformed into itself by every isomorphism of G, is called^ a 
 characteristic sub-group of G. 
 
 A characteristic sub-group of a group G is necessarily a 
 self-conjugate sub-group of G ; but a self-conjugate sub-group 
 is not necessarily characteristic. A simple group, having no 
 self-conjugate sub-groups, can have no characteristic sub- 
 groups. Let now G be any group, and let K be the holomorph 
 of G. A characteristic sub-group of G is then a self-conjugate 
 sub-group of K; and conversely, every self-conjugate sub-group 
 of K which is contained in C is a characteristic sub-group of G. 
 
 Suppose now a chief-series of K formed which contains 
 G. If G has no characteristic sub-group, it must be the 
 last term but one of this series, the last term being identity. 
 It follows by § 94 that G must be the direct product of a 
 number of simply isomorphic simple groups. Hence : — 
 
 Theorem IV. A group, which has no characteristic suh,- 
 group, must he either a simple group or the direct product of 
 simply isomorphic simple groups. 
 
 The converse of this theorem is clearly true. 
 
 163. Suppose now that G^ is a group which has character- 
 istic sub-groups ; and let 
 
 ") Gi, , Gr, Gr+i, ) 1 
 
 be a series of such sub-groups, each containing all that follow it 
 and chosen so that, for each consecutive pair G^and G'r+i, there 
 is no characteristic sub-group of G contained in Gr and con- 
 taining Gr+1, except G^+i itself Such a series is called'' a 
 characteristic series of G. 
 
 1 Frobenius, "Ueber eudliohe Gruppen," Berliner Sitzungsberichte, 1895, 
 p. 183. 
 
 2 Frobenius, "Ueber auflbsbare GruTppen, n," Berliner Sitzungsberichte, 1895, 
 p. 1027. 
 
164] CHARACTERISTIC SERIES 233 
 
 It may clearly be possible to choose such a series in more 
 than one way. If 
 
 ^> "1, , (Xr > 'JT r-n> ) 1 
 
 be a second characteristic series of Q, then 
 
 K, J, , H, G, Gi, , Qr, Or+i, , 1 
 
 and K, J, , H, 0, G,', , G/, G'r+u , 1 
 
 are two chief-series of K. In fact, if K had a self-conjugate 
 sub-group contained in Gr and containing Gr+i, then G would 
 have a characteristic sub-group contained in Gr and containing 
 Gr+i. The two chief-series of K coincide in the terms from K 
 to G inclusive. Hence the two sets of factor-groups 
 
 ■ G G^ G^ 
 
 Gi' G2' ' Gr+i' 
 
 J G Gi Gr 
 
 must be equal in number and, except possibly as regards the 
 sequence in which they occur, identical in type. Moreover, each 
 factor-group must be either a simple group or the direct product 
 of simply isomorphic simple groups. 
 
 164. We will now shew how to determine a characteristic 
 series for a group whose order is the power of a primed 
 
 First, let the group G be Abelian ; and suppose that it is 
 generated by a set of independent operations, of which Wj are of 
 the order p^-, (s = 1, 2, . . ., r), while 
 
 ?7ii >m2 > >my. 
 
 The sub-group G^ (§ 42), formed of the operations of G 
 which satisfy the relation 
 
 is clearly a characteristic sub-group. As a first step towards 
 forming the characteristic series, we may take the set of 
 groups 
 
 Gmt\= G), Gm^—i, Gm^—i, > G^, Cti, Ij 
 
 for this is a set of characteristic sub-groups such that each 
 contains the one that follows it. 
 
 1 Frobenius, I.e., pp. 1028, 1029. 
 
234 CHARACTERISTIC [164 
 
 Now the sub-group H„ (§ 45), formed of the distinct 
 operations that remain when every operation of Q is raised 
 to the power p", is also a characteristic sub-group ; and since 
 the operations common to two characteristic sub-groups also 
 form a characteristic sub-group, the sub-group K^^ y (common to 
 Of^ and Hy) is characteristic. It follows from this that G^ will be 
 a characteristic sub-group only when r = 1. If r > 1, (tj is not 
 contained in Hm_-i, and the common sub-group K^ „ _i of 
 these two is characteristic. If r > 2, this sub-group again is 
 not contained in Hm^^-i ', and the common sub-group K^^ ^ _i 
 of Gi and Hm^_^-i is a characteristic sub-group contained in 
 ^i,»K.i-i- Continuing thus, we form between G^ and 1 the 
 series 
 
 U'l, -"■i,))!.,_i— 1) -"-i.m^j— 1> , -lii.mt,— 1, 1. 
 
 In a similar way, between (r„ and (?„_! we introduce such of 
 the series 
 
 as are distinct, the symbol m, — a being replaced by zero where 
 it is negative. 
 
 From the original series we thus form a new one, in which 
 again each group is characteristic and contains the following. 
 This series may be shewn to be a characteristic series. 
 
 be the «» generating operations of G, whose orders are p™: 
 Then if {Ga-i, K^^m,-o] and [G0.-1, K^^m^^-o\ are distinct, the 
 generating operations of the latter differ only from those of the 
 former in containing the set 
 
 in the place of 
 
 »,-« (^^12, , «,). 
 
 m., a ' ^ ' ' ' »/ 
 
 Now any permutation of the Us generating operations 
 
 Pm„x, («=1, 2, , lis), 
 
 among themselves, the remaining generating operations being 
 unaltered, must clearly give an isomorphism of G with itself; 
 and therefore no sub-group of (r, contained in [Ga-i, Ka,m-ti\ 
 
165] SERIES 235 
 
 and containing {Ga-i, J^a,m,_^-a], can be a characteristic sub- 
 group. This result being true for every pair of distinct groups 
 which succeed each other in the series that has been formed, 
 it follows that the series is a characteristic series. It may be 
 noticed that, if F and V are any two consecutive sub-groups in 
 
 r 
 
 a characteristic series of G, the order of =^, must be p", where v 
 is one of the r numbers ng. 
 
 Secondly, suppose that G is not Abelian. We may first 
 consider the series of sub-groups 
 
 of § 53. Each of these is clearly a characteristic sub-group, and 
 
 rr 
 
 each contains the succeeding. Moreover, -~^ is an Abelian 
 
 group ; and, by the process that we have just investigated, a 
 characteristic series may be formed for it. To each group in 
 this series will correspond a characteristic sub-group of if^+i 
 containing Hr', and the set of groups so obtained forms part of 
 a complete characteristic series of G. When between each 
 consecutive pair of groups in the above series the groups thus 
 formed are interpolated, the resulting series of groups is a 
 characteristic series for G. 
 
 165. The isomorphisms of a given group with itself are 
 closely connected with the composition of every composite group 
 in which the given group enters as a self-conjugate sub-group. 
 Let G be any composite group and H a self-conjugate sub- 
 group of G. Then since every operation of G transforms H 
 into itself, to every such operation will correspond an iso- 
 morphism of H with itself If 8 is an operation of G not 
 contained in H, and if the isomorphism of H arising on 
 transforming its operations by S is contragredient, so also is 
 the isomorphism arising from each of the set of operations SH. 
 In this case, no one of this set of operations is permutable with 
 every operation of H. If however the isomorphism arising from 
 S is cogredient, there must be some operation h oi H which 
 gives the same isomorphism as 8 ; and then Sh'^ is permutable 
 with every operation of H. In this case, the set of operations 
 8H will give all the cogredient isomorphisms of H. 
 
236 COMPLETE [165 
 
 Suppose now that -ffj is that sub-group of which is formed 
 of all the operations of Q that are permutable with every 
 operation of H. Then to every operation of 0, not contained 
 in [H, Hi], must correspond a contragredient isomorphism of 
 
 H; and to every operation of the factor group , corre- 
 
 spends a class of contragredient isomorphisms. If then L is the 
 group of isomorphisms of H, and if L^ is that self-conjugate 
 sub-group of L which gives the cogredient isomorphisms of H, 
 
 C T 
 
 ,^, must be simply isomorphic with a sub-group of -j^ . 
 
 If now H contains no self-conjugate operation except 
 identity, H and H^ can contain no common operation except 
 identity ; and since each of them is a self-conjugate sub-group 
 of G, every operation of H is permutable with every operation 
 of Hi. In this case, {H, H^ is the direct product of H and H^. 
 
 If, further, L coincides with H, so that H admits of no 
 
 C 
 contragredient isomorphisms, ,„ „, must reduce to identity. 
 
 In this case, is the direct product of H and Hi. 
 
 Definition. A group, which contains no self-conjugate 
 operation except identity and which admits of no contra- 
 gredient isomorphism, is called^ a complete group. 
 
 The result of the present paragraph may be expressed in 
 the form : — 
 
 Theorem V. A group, which contains a complete group as 
 a self-conjugate sub-group, must he the direct product of the 
 complete group and some other group''- 
 
 166. Theorem VI. If is a group with no self-conjugate 
 operations except identity; and if the group of cogredient iso- 
 morphisms of G is a characteristic sub-group of L, the group of 
 isomorphisms of G ; then L is a complete groups. 
 
 ' Holder, "Bildung zusammengesetzter Gruppen,'' Math. Ann., Vol. xlvi 
 (1895), p. 325. 
 
 ^ Ibid. p. 325. 
 
 ' Holder (loc. cit. p. 331) gives a theorem which is similar but not quite 
 equivalent to Theorem VI. 
 
166] GROUPS 237 
 
 With the notation of § 158, the operations of L may be 
 represented by the substitutions 
 
 ©■ 
 
 -G 
 
 The group of cogredieat isomorphisms, which we will call G', is 
 given by the substitutions 
 
 U4j=(-=^'i -^-1)' 
 
 it is simply isomorphic with 0. 
 
 and therefore no operation of L is permutable with every 
 operation of G'. Hence every isomorphism of G' is given on 
 transforming its operations by those of L. Suppose now 
 that J is an operation which transforms L into itself. Since 
 G' is by supposition a characteristic sub-group of L, the 
 operation / transforms G' into itself If / does not belong to 
 L, we may assume that J is permutable with every operation of 
 G'. For if it is not, it must give the same isomorphism of G' 
 as some operation 8 oi L; and then JS~^ is permutable with 
 every operation of G', and is not contained in L. Now / being 
 permutable with every operation of G', we have 
 
 J~^s~^gsJ = s~^gs, 
 
 where s is any operation of L, and g any operation of G'. 
 
 Moreover JgJ~^ = g, 
 
 and therefore 
 
 J~^s~^JgJ~^sJ = s~^gs. 
 
 Hence s and J~^sJ give the same isomorphism of G' Now 
 no two distinct operations of L give the same isomorphism of 
 G', so that s and J~^sJ must be identical; in other words, J 
 is permutable with every operation of L. Hence L admits of 
 no contragredient isomorphisms. Moreover, G' has no self- 
 conjugate operations, and no operation of L is permutable with 
 
238 COMPLETE [166 
 
 every operation of G'; hence L has no self-conjugate operations. 
 It is therefore a complete group. 
 
 Corollary. If (? is a simple group of composite order, or 
 if it is the direct product of a number of isomorphic simple 
 groups of composite order, the group of isomorphisms i of (? is 
 a complete group. 
 
 For suppose, if possible, in this case that Q' is not a 
 characteristic sub-group of L ; and that, by a contragredient 
 isomorphism of L, G is transformed into G". Then G" is a 
 self-conjugate sub-group of L, and each of the groups G' and 
 G" transforms the other into itself Hence (§ 34) either every 
 operation of G' is permutable with every operation of G", or 0' 
 and G" must have a common sub-group. The former supposi- 
 tion is impossible since no operation of L is permutable with 
 every operation of G. On the other hand, if G' and G" have a 
 common sub-group, it is a self- conjugate sub-group of L and it 
 therefore is a characteristic sub-group of G'. Now (§ 162) G has 
 no characteristic sub-groups, and therefore the second supposition 
 is also impossible. It follows that, in this case, G' is a character- 
 istic sub-group of L, and that L is a complete group. 
 
 167. Theorem VII. If G is an Abelian group of odd order, 
 and if K is the holomorph of G ; then when G is a characteristic 
 sub-group of K, the latter group is a complete group. 
 
 If N is the order of G, then K can be expressed (§ 158) as a 
 transitive group of degree N. When K is so expressed, those 
 operations of K which leave one symbol unchanged form a 
 sub-group H, which is simply isomorphic with the group of 
 isomorphisms of G. Now (§ 160) an Abelian group of odd order 
 admits of a single isomorphism of order two, which changes every 
 operation into its own inverse. The corresponding substitution 
 of .ff is a self-conjugate substitution in H, and is one of N con- 
 jugate substitutions in K. These are the only substitutions 
 of K which transform every substitution of G into its inverse. 
 If ff is a characteristic sub-group of K, every isomorphism of K 
 must transform G, and therefore also the set of N conjugate 
 substitutions of order two, into itself Also, no substitution of 
 K can be permutable with each one of these N substitutions, 
 since each of them keeps just one symbol unchanged. Hence 
 
168] GROUPS 239 
 
 (Theorem III, Cor. II, § 161) the group of isomorphisms of K 
 can be expressed as a transitive group of degree N, which 
 contains G* as a transitive self-conjugate sub-group. But 
 when the group of isomorphisms of K is so expressed, K itself 
 consists of all the substitutions of the N symbols which are 
 permutable with 0; and at the same time, every isomorphism 
 of K transforms G into itself Hence the group of isomorph- 
 isms of K must coincide with K itself; i.e. K admits of no 
 contragredient isomorphisms. Also K obviously contains no 
 self-conjugate operation except identity ; hence it is a complete 
 group. 
 
 Corollary. The holomorph of an Abelian group of order 
 p^, where p is an odd prime, and type (1, 1, ... to m units), is a 
 complete group. For if, in this case, Q is not a characteristic 
 sub-group of K, let G' be a sub-group of K which, in the group 
 of isomorphisms of K, is conjugate to G. Then G and G' being 
 both self-conjugate in K must have a common sub-group, since 
 K cannot contain their direct product. But the common sub- 
 group of G and G', being self-conjugate in K, is a characteristic 
 sub-group of G. This is impossible (§ 162); hence (? is a 
 characteristic sub-group of K. 
 
 Ex. Shew that the holomorph of an Abelian group of degree 
 2™ and type (1, 1, ... to m units) is a complete group. 
 
 168. We shall now discuss the groups of isomorphisms of 
 certain special groups ; and we shall begin with the case of a 
 cyclical group 0, of prime order p, generated by an operation P. 
 Every isomorphism of such a group must interchange among them- 
 selves the ^ — 1 operations 
 
 and if any isomorphism replaces P by P", it must replace P"^ by P^", 
 and so on. Moreover, the symbol 
 
 / P P" PP-' \ 
 
 ( pap2a p[p-l)aj 
 
 does actually represent an isomorphism whatever number a may be 
 from 1 to p -1 ; for each operation occurs once in the second line, 
 and the change indicated leaves the multiplication-table of the group 
 unaltered. If a =p, the symbol does not represent an isomorphism : 
 ii a-p + a, the symbol represents the same isomorphism as 
 p pi pp-i \ 
 
 po.1 pia' p{p-l)aL'J • 
 
240 GROUP OF ISOMORPHISMS [168 
 
 The group of isomorphisms of a group of prime order p is 
 therefore a group of order p-\. Now the nth power of the 
 isomorphism 
 
 '" (/■)■ 
 
 Hence if a is a primitive root of the congruence 
 
 aP-i -1 = 0, (mod. p), 
 
 the group of isomorphisms is a cyclical group generated by the 
 isomorphism 
 
 {;■)■ 
 
 Finally, if ,5^ is an operation satisfying the relations 
 
 where a is a primitive root of p, {S, P} is the holomorph of G. 
 
 The reader will at once observe that this group of order p(p-l) 
 is identical with the doubly transitive group of § 112. It is a 
 complete group. 
 
 169. We shall consider next the case of any cyclical group. 
 
 Suppose, first, that ff is a cyclical group of order p", where p is 
 an odd prime ; and let it be generated by an operation &'. The group 
 contains p^~^ (p - I) operations of order p" ; if S' is any one of these. 
 
 (?) 
 
 defines a distinct isomorphism. The group of isomorphisms is there- 
 fore a group of order p^~^{p — 1). Moreover, since the congruence 
 ^p'-\p-i) _ 1 = 0, (mod. p"), 
 
 has primitive roots, the group of isomorphisms is a cyclical group. 
 The holomorph of Gf is defined by 
 
 ^^" = 1, /i'"-Mi'-i)=l, J-^SJ=S% 
 
 where a is a primitive root of the congruence 
 
 ^p"-i(p-i)_l = 0, (mod. ^"). 
 
 If G is a, cyclical group of order 2^, it follows, in the same way, 
 that the group of isomorphisms is an Abelian group of order 2""^ 
 In this case, however, the congruence 
 
 of-' - 1 B 0, (mod. 2"), n>2, 
 has no primitive root, and therefore the group of isomorphisms is not 
 cyclical. The congruence 
 
 of""' -1 = (mod. 2") 
 
170] OF A CYCLICAL GROUP 241 
 
 however has primitive roots, and a primitive root a of this con- 
 gruence can always be found to satisfy the condition 
 
 a=""' = 1 + 2»-S (mod. 2"). 
 
 The powers of the isomorphism 
 
 Q-) 
 
 then form a cyclical group of order 2" ^ ; and the only isomorphism 
 of order 2 contained in it is 
 
 Hence Q and (/_,) , 
 
 the latter not being contained in the sub-group generated by the 
 former, are two permutable and independent isomorphisms of orders 
 2""^ and 2. They generate an Abelian group of order 2""', which is 
 the group of isomorphisms of G. The corresponding holomorph is 
 given by 
 
 e/j" loJi = o , J ^oJ <^ = o ~^, 
 
 where a satisfies the conditions given above. 
 
 If G^ is a cyclical group of order 4, its group of isomorphisms is 
 clearly a group of order 2. 
 
 170. It is now easy to construct the group of isomorphisms of 
 any cyclical group G, and the corresponding holomorph. If the order 
 
 of G is '2"pi'^ip™i , where p^, p^, are odd primes, G is the 
 
 direct product of cyclical groups of orders 2", 2h'^', P™h ; and 
 
 every isomorphism of G transforms each of these groups into itself. 
 Hence if the groups of isomorphisms of these cyclical groups be 
 formed, and their direct product be then constructed, every operation 
 of the group so formed will give a distinct isomorphism of the group 
 G. Moreover, the order 
 
 2»->™.-i(^i-l)p™.-i(^,-l) 
 
 of this group is equal to the number of operations of G whose order 
 
 is 2"pi™ip2™2 or in other words to the number of isomorphisms 
 
 of which G is capable. The group thus formed is therefore the group 
 of isomorphisms of G. The corresponding holomorph is clearly the 
 direct product of the holomorphs of the cyclical groups of orders 2", 
 ji;i™i, etc. 
 
 When the order of G is odd, the holomorph K is easily shewn 
 to be a complete group. Suppose it to be expressed transitively, 
 as in § 158, in N symbols, where N is the order of ff ; if 6^ is 
 not a characteristic sub-group of K, let G" be a group into which 
 G is transformed by a contragredient isomorphism of K. Then G' 
 
 B. 16 
 
242 THE LINEAR [170 
 
 is a self-conjugate sub-group of K ; and since G' is cyclical, every 
 sub-group of G' is a self -conjugate sub-group of K. Hence a 
 generating operation of G' must be a circular substitution of 
 iSf symbols. The N operations of order 2 which transform each 
 operation of G' into its own inverse therefore each keep one symbol 
 fixed ; hence each of them must transform every operation of G 
 into its inverse. But there is only one such set of operations 
 of order 2, and therefore G' cannot differ from G. It follows by 
 Theorem VII, § 1 67 that, as G is a characteristic sub-group of K the 
 group K itself is complete. 
 
 If the order of G is even, K must contain a self-conjugate 
 operation other than identity, namely the operation of order 2 
 contained in G. Moreover, K admits of a contragredient isomorph- 
 ism whose square is cogredient. From the mode of formation of 
 K, it is clearly sufficient to verify this when the order of 6^^ is a 
 power of 2. The holomorph K is given by 
 
 5^"=1, J/-'=l, Ji=l, J^J„^ = J^J„ 
 
 where a is a primitive root of 
 
 a^"-'-] =0 (mod. 2»), 
 such that o?"~' +1^0 (mod. 2"). 
 
 In this group, J^ is one of 2""' conjugate operations 
 
 2) *'2'^ ) t/2*J , 
 
 Now it may be directly verified that K admits the isomorphism 
 represented by 
 
 Moreover, since this isomorphism changes J^ into J^S, it cannot be a 
 cogredient isomorphism. Finally, the square of this isomorphism is 
 
 {0, Ji , J2 
 
 \S, J,S'-% J,S' J' 
 
 /S, Jl , ^2 \ 
 
 which is a cogredient isomorphism. 
 
 ' 171. We shall next consider the group of isomorphisms of an 
 Abelian group of order jo" and type (1, 1, ... to n units). Such a 
 group is generated by n independent permutable operations of 
 order p, say 
 
 P P P 
 
171] HOMOGENEOUS GROUP 243 
 
 Since every opei-ation of the group is self-conjugate and of order 
 p, while the group contains no characteristic sufe-group, there must 
 be isomorphisms transforming any one operation of the group into 
 any other. We may therefore begin by determining under what 
 conditions the symbol 
 
 defines an isomorphism. This symbol replaces the operation 
 Pf'P^''' P/" by P/.P/» P/", where 
 
 
 (mod. p). 
 
 Unless the p^ operations P/i P/z P/" thus formed are all 
 
 distinct, when for P-^^Pfi P^" is put successively each of the p" 
 
 operations of the group, the symbol does not represent an isomorphism. 
 On the other hand, when this condition is satisfied, the symbol 
 represents a permutation of the operations among themselves which 
 leaves the multiplication table of the group unchanged ; it is there- 
 fore an isomorphism. 
 
 If this condition is satisfied, x-^, x^, , x^ must be definite 
 
 numbers (mod. p), when y^, y^, , «/„ are given; and therefore 
 
 the above set of n simultaneous congruences must be capable 
 of definite solution with respect to the k's. The necessary and 
 sufficient condition for this is that the determinant 
 
 «21I 
 
 ai2, ••■ 
 
 a^, ... 
 
 
 »M, 
 
 a„2j ■ • • 
 
 ••, «»m 
 
 should not be congruent to zero (mod. p). 
 
 Every distinct set of congruences of the above form, for 
 which this condition is satisfied, represents a distinct isomorph- 
 ism of the group, two sets being regarded as distinct if the 
 congruence 
 
 a^, = «V» (mod. p) 
 
 does not hold for each corresponding pair of coefficients. Moreover, 
 to the product of two isomorphisms will correspond the set of con- 
 gruences which results from carrying out successively the operations 
 indicated by the two sets that correspond to the two isomorphisms. 
 
 16—2 
 
244 
 
 THE LINEAE 
 
 [171 
 
 The group of isomorphisms is therefore simply isomorphic with 
 the group of operations defined by all sets of congruences 
 
 
 ' (mod. p) 
 
 Vn — ^711 ^1 + '^7i2'^2 "^ "^ ^nn ^n 
 
 for which the relation 
 
 is satisfied. 
 
 ttu , «!, 
 
 •) ^2n 
 
 •> "-17! 
 
 ^ (mod. p) 
 
 '^21! 
 
 flSlo, ... 
 
 «.„, ... 
 
 
 ^r\ J 
 
 a,-2, ■•■ 
 
 • • ! ''it 
 
 172. The group thus defined is of great importance in many 
 branches of analysis. It is known as the linear homogeneous group. 
 In a subsequent Chapter we shall consider some of its more 
 important properties ; we may here conveniently determine its 
 order. Let this be represented by N^, so that N^ represents the 
 number of distinct solutions of the congruences 
 
 ^ (mod. p). 
 
 Since the group of isomorphisms of the Abelian group of order 
 7;" transforms every one of its operations (identity excepted) into 
 every other, it can be represented as a transitive substitution group 
 of «" — 1 symbols, and therefore, if M is the order of the sub-group 
 that keeps Pj unchanged, 
 
 N^ = {p^-\)M. 
 Now, in the congruences corresponding to an operation that keeps 
 f, unchanged, we have 
 
 fflii^l, ai2Sai3= =a^n = Q, (mod. p). 
 
 Hence M is equal to the number of solutions of the congruences 
 1, 0, ,0 
 
 ^711 ) ^712 7 7 ^71 
 
 ^ (mod. p). 
 
 The value of the determinant does not depend on the values of 
 , a.,], , «„i, and therefore 
 
173] HOMOGENEOUS GROUP 245 
 
 Hence N^ = {p"- - \)f-''N^_.„ 
 
 and therefore immediately 
 
 Nn = {p''-\){f'-p) (p^-p™-!). 
 
 The reader will notice that an independent proof of this result 
 has already been obtained in § 48. The discussion there given of 
 the number of distinct ways, in which independent generating 
 
 operations of an Abelian group of type (1, 1, , 1) may be 
 
 chosen, is clearly equivalent to a determination of the order of the 
 corresponding group of isomorphisms. 
 
 The holomorph of an Abelian group, of order p" and type 
 (1, 1, ... to n units), can similarly be represented as a group of 
 linear transformations to the prime modulus 2). Consider, in fact, 
 the set of transformations 
 
 yi = »na=i +»i2a;2 + + a»a;„ +6j, 
 
 2/2 = a^^x^ + a^^x^ + + a^^x^ + b^, 
 
 (mod. p) ; 
 
 Vn ~ a^iXi + a^jKj + + a^nXn + K, 
 
 where the coefficients take all integral values (mod. p) consistent 
 
 with 
 
 $0. 
 
 The set of transformations clearly forms a group whose order is 
 N„p^'. The sub-group formed by all the transformations 
 
 2/iSa;i+6i, y^ = x^ + b^, , y^ = x^ + h^, (mod. p), 
 
 is an Abelian group of order/*" and type (1, 1, ... to to units), and it 
 is a self -conjugate sub-group. Moreover, the only operations of the 
 group, which are permutable with every operation of this self-con- 
 jugate sub-group, are the operations of the sub-group itself; and, 
 since the order of the group is equal to the order of the holomorph 
 of the Abelian group, it follows that the group of transformations 
 must be simply isomorphic with the holomorph of the Abelian 
 group. In the simplest instance, where p^ is 2^, the holomorph is 
 simply isomorphic with the alternating group of four symbols. In 
 any case the holomorph, when"'6kpressed as in § 158, is a doubly 
 transitive group of degree p". 
 
 173. It has been seen in § 142 that, except when n = 6, the 
 symmetric group of degree ri has n and only ?i sub-groups 
 of order n — l\, which form a conjugate set. Hence by 
 Theorem III, § 161, the group of isomorphisms of the sym- 
 metric group of degree n can be expressed, except when n=6, 
 
246 FURTHER GROUPS [173 
 
 as a transitive group of degree n. The symmetric group of n 
 symbols however consists of all possible substitutions that can 
 be performed on the n symbols, and therefore it must coincide 
 with its group of isomorphisms. Hence^ : — 
 
 Theorem VIII. The symmetric group of n symbols is a 
 complete group, except when n = 6. 
 
 Corollary. Except when w = 6, the alternating group of n 
 symbols admits of one and only one class of contragredient 
 isomorphisms. 
 
 For with this exception, the alternating group of degree n 
 has just n sub-groups of order ^ (w — 1)!. 
 
 174. The alternating group of degree 6 occurs as a special 
 case of another class of groups of which we will determine the 
 groups of isomorphisms. These are the doubly and the triply transi- 
 tive groups that have been defined in §§ 11 2, 1 1 3. 
 
 The doubly transitive group of degree p^ and order p'^{p™—\) 
 there considered has a single set of p™ conjugate cyclical sub-groups 
 of order p™—\. Its group of isomorphisms can therefore be ex- 
 pressed as a transitive group of degree jo'". Let p^ [p™ - 1) ju be the 
 order of the group of isomorphisms. The order of a sub-group that 
 keeps two symbols fixed is /«, ; and every operation of this sub-group 
 must transform a cyclical sub-group of order p^—l into itself. With 
 the notation of § 1 1 2, we will consider the sub-group which keeps 
 and i fixed. Every operation of this sub-group must transform the 
 cyclical sub-group generated by the congruence^ 
 
 x = ix 
 
 into itself ; and no operation of the sub-group can be permutable 
 with the given operation. If S is an operation of the sub-group 
 which transforms 
 
 x' = ix 
 
 into oa' = i^x, 
 
 then S, when represented as a substitution, is given by 
 
 / ^^ i', i\ \ 
 
 Vi'+S ■i^'^+i, ^^''+^ /■ 
 
 Now S must transform the sub-group of order ^"* into itself ; it 
 must therefore be permutable with that operation of this sub-group 
 which changes into i. 
 
 1 Holder, Math. Ann. Vol. xlvi, (1895), p. 343. 
 
 ' This and all subsequent congruences in the present section are to be taken, 
 mod. p. 
 
174] OF ISOMORPHISMS 247 
 
 This operation is given by 
 
 x' = x + i, 
 
 and if we denote it by T, then S~'^TS changes i^+^ into i'^^^, where 
 
 y 
 i" ^^ + i = i^*'^. 
 
 Now T changes V*'^ into i"*^ + i. Hence, since S and T are 
 permu table, we must for all values of y have the simultaneous 
 congruences 
 
 and i"*^ + i = ■i''^+'- 
 
 Eliminating z, the congruence 
 
 V 
 
 \+ii = {l+i'>-y 
 
 must hold for all values of y from top'"— 1. If a is not a power 
 of p, this congruence involves an identity of the form 
 
 ait"»i + a^i"^ + =0, 
 
 where all the indices are less than jij™ - 1 ; and this is impossible. 
 
 Hence the only possible values of a are I>! P'> P'j > ^^^ the 
 
 greatest possible value of /u is m. 
 
 Now the congruence 
 
 x' = x'' 
 
 defines a substitution performed on the p™ symbols permuted by the 
 group, and this substitution is permutable with the group. For if 
 we denote this operation by J, and any operation 
 
 x' = ax + ^ 
 of the group by 2, then J~^%J is 
 
 x' = aPx + yS^, 
 
 another operation of the group. Moreover, J clearly transforms 
 
 «' = ix 
 into its |)th power. 
 
 The group of isomorphisms of the doubly transitive group of 
 order p™ is therefore the group defined by 
 
 iAj 1^. l/JUj yC ^z vC "r" i/m 3(j ^; uC , 
 
 where i is a primitive root of the congruence 
 
 l'^ = 1. 
 
 From this it immediately follows that the group of isomorphisms 
 of the triply transitive group, of degree p™ + 1 and order 
 
248 FUETHER GROUPS [174 
 
 defined by the congruences 
 
 , _ax + fi 
 
 where a, /3, y, S satisfy the conditions of § 113, is the group of order 
 (p'"+ l)^™(p"'-l)m obtained by combining the previous con- 
 gruences with 
 
 «' = xo. 
 
 In fact, it may be immediately verified that the operation given 
 by this congruence is permutable with the group, and does not give 
 a cogredient isomorphism of the group. Moreover, by Theorem III, 
 § 161, the group of isomorphisms of the given group can be expressed 
 as a transitive group of degree p" + 1 ; and therefore, among a class of 
 contragredient isomorphisms, there must be some transforming into 
 itself a sub-group which keeps one symbol fixed. Hence the order 
 of the group of isomorphisms cannot exceed 
 
 When J) is an odd prime, the triply transitive group of degree 
 ^"' + 1, which may be defined by 
 
 1 , . , 
 
 K— , X =%X, X =X + l, 
 
 X ' 
 
 contains as a self -conjugate sub-group a doubly transitive group 
 defined by 
 
 X = X = %''X, X ^ X+ I. 
 
 X 
 
 It will be shewn in Chapter XIV that this is a simple group. 
 When p^ is equal to 3^, it is easy to verify that this group is simply 
 isomorphic with the alternating group of degree 6. 
 
 The group of isomorphisms of this simple group can be expressed 
 as a transitive group of degree ;j"' -i- 1, and must clearly contain the 
 triply transitive group of order (jb™-i- l)jo™(p"'- 1) self-conjugately. 
 It therefore coincides with the group of order (p'" -y 1) p™ (p™- 1) to, 
 which has just been determined. The latter group is therefore 
 (Theorem VI, Cor. § 166) a complete group. Further, if the simple 
 group be denoted by G and the group of isomorphisms by L, the 
 
 factor group -^, will be determined by 
 
 tAr tilj^ tAj ^— iAj , 
 
 when all operations of G are treated as the identical operation. 
 Denoting these operations by / and J, then J~^IJ is /*, which is 
 
 the same as [ multiplied by an operation of G. Hence -=; is an 
 
 Abelian group generated by two independent operations of orders 2 
 and m. 
 
175] OF ISOMORPHISMS 249 
 
 The group of isomorphisms of the alternating group of degree 6 
 is therefore a group of order 1440; and the symmetric group of 
 degree 6 admits a single class of contragredient isomorphisms^. 
 
 175. Let P be any group whose order is the power of a 
 prime, and let 
 
 of orders p", p"', p°-', , j)"", 1 
 
 be a characteristic series (§ 163) of P. Every isomorphism 
 of P must transform P^ and Pf+i into themselves, and there- 
 
 P 
 
 fore also p-^ into itself Suppose now that an isomorphism 
 
 7 of P transforms every operation of Pr+i into itself and 
 
 P . . 
 
 every operation of p-^ into itself If 8 is any operation of 
 
 Pr, not contained in P^+i, / must transform 8 into SA, where 
 A is some operation of Pr+i ', so that, if p*^ is the order of 
 A, I transforms the operations of the set SP^+i cyclically 
 among themselves in sets oip^^. Similarly, if fif' is an operation 
 of P, not contained in the set SPr+i, I will transform the 
 operations of the set S'Pr+i cyclically among themselves in 
 sets of ^'''- Hence the order of the isomorphism / is a multiple 
 of p ; and any isomorphism, that transforms every operation 
 
 Pr 
 
 of P,.+i into itself and every operation of -p-^ into itself and is 
 
 of order prime to p, will transform every operation of P,. into 
 itself Therefore, the only isomorphism of P, that transforms 
 every operation of each of the groups 
 
 ■t^ -T 1 -t «— 1 p 
 
 > "p" > ' p ' -* « 
 
 1 -'2 -* m 
 
 into itself and is of order prime to p, is the identical isomorph- 
 ism. Now each of these groups is an Abelian group, whose 
 operations are all of order p ; and it has been shewn that, 
 if j)'" is the order of such a group, the order of its group of 
 isomorphisms is 
 
 (^m _ 1 -) (-^771 _ p) (p'«- - p"'-l). 
 
 I Holder, loc. cit. p. 343. 
 
250 ORDERS OF THE ISOMORPHISMS [175 
 
 Every isomorphism of P, whose order is relatively prime to 
 p, must therefore be such that its order is a factor of one of the 
 expressions of the above form, obtained by writing 
 
 «-«!, ai-«2, , On, 
 
 in succession for m. If then k is the greatest of these numbers, 
 the order of any isomorphism of P, whose order is relatively 
 prime to p, is a factor of 
 
 ip-l)(p'-l) (p^-l). 
 
 Herr Frobenius^ has introduced the symbol ^(P) to denote 
 
 this product. If (? is a group of order pi'^p^'' ^„*", where 
 
 Pi,P2, ,Pn are distinct primes, and if P^, P^ ,P„are 
 
 groups of orders ^I's p/^ , ^„«« contained in G, we shall use 
 
 the symbol ^{G) to denote the least common multiple of 
 
 176. If P' is a sub-group of a group P of order p", ^(P') 
 is not necessarily a factor of ^ (P). For instance, the group of 
 order ^^ generated by the four operations A, B, G, D of order ^, 
 of which A and B are self-conjugate while 
 
 D-'CD = GA, 
 
 has a characteristic sub-group \A, B] of order p", and it 
 has no characteristic sub-group of order p^. Hence 
 
 ^{P) = (p-l)(f-l). 
 
 The sub-group P' however, which is generated by A, B and 
 G, is an Abelian group whose operations are all of order p ; and 
 therefore 
 
 ^(P') = (p-l)(p^-l){f-l). 
 
 So again, for the Abelian group P of order p', generated by 
 A and B, where 
 
 AP'=1, Bp = 1, AB = BA, 
 
 we have ^ (P) = ^ — 1 ; 
 
 while its sub-group P', generated by A^ and B, is an Abelian 
 group whose operations are all of order p, so that 
 
 ^(P') = (p-l)(p^-l). 
 
 ' " Ueber auflosbare Gruppen, ii," Berliner Sitzungsberichte, 1895, p. 1028. 
 
177] OF A GIVEN GROUP 251 
 
 Suppose now that P is an Abelian group of order p", 
 generated by p independent and permutable operations. For 
 such a group, we define' a new symbol (P) by the equation 
 
 ^(P) = (p-1)(^^-1) (p-'-l). 
 
 In forming the characteristic series of P in § 164, we 
 
 commenced with the series of groups Gr (r = l, 2, ), such 
 
 that Or consists of all the operations of P satisfying the 
 relation 
 
 SP' = 1. 
 
 Since P is generated by p independent operations, the order 
 
 of Gi is pi', and the order of -^ cannot be greater than p''. If 
 
 the generating operations of P are all of the same order /8, the 
 series 
 
 P, Gfp-i, Gffi-2, , Gi, 1 
 
 is a complete characteristic series of P, and each factor-group 
 -p^ is of order pi". In this case, therefore, 
 
 ^(P) = d(P). 
 
 If however the generating operations are not all of the same 
 order, Gi will not be the last term of the complete characteristic 
 series ; nor will ^,.+1 and Gr be consecutive terms in the series, if 
 
 the order of -^ is pi". Hence, in this case, ^ (P) will be a 
 
 factor of 9(P). 
 
 If now P' is any sub-group of P, then it has been seen 
 (§ 46) that the number of independent generating operations of 
 P' is equal to or less than p. Hence 0{P') is equal to or is a 
 factor of ^(P). 
 
 177. Theorem IX. If a group G, of order N, is trans- 
 formed into itself by an operation S, whose order is relatively 
 prime to N^ (G), every operation of G is permutable with S*. 
 
 Let Pj"' be the highest power of a prime p^ which divides N. 
 
 1 Frobenius, loc. cit. p. 1030. 
 * Frobenius, loc. cit. p. 1030. 
 
252 ORDERS OF THE ISOMORPHISMS [17^ 
 
 The number of sub-groups of 0, and therefore also of {,S^, (}\ 
 whose order is pi«', is a factor of N; and since the order of 's is 
 relatively prime to N, one at least of them, say P^, must be 
 transformed into itself by S. Now the order of S is relatively 
 prime both to p^ and to ^ (Pj) ; and therefore the isomorphism of 
 P given on transforming its operations by S must be the 
 identical isomorphism. In other words, 8 is permutable with 
 every operation of Pj. In the same way it follows that, if 
 
 i^/S p/S are the highest powers oi p^, p„ that divide 
 
 JSf, there must be sub-groups of orders p.f\ pj^', with every 
 
 operation of each of which S is permutable. But these groups 
 of orders Pi"', p2''\ Ps"', generate G ; therefore *Sf is permu- 
 table with every operation of G. 
 
 Corollary. If the order of an isomorphism of G is rela- 
 tively prime to N, it must be a factor of ^ (G). 
 
 It follows immediately, from the theorem, that there is no 
 isomorphism of G whose order contains a prime factor not 
 occurring in N^(G). Suppose, if possible, that the group of 
 isomorphisms of G contains an operation S of order g'", where 
 g is a prime which does not divide N, and that the highest 
 power of q that divides ^(G) is q™', where m'< m. Ifjj/r is 
 the highest power of any prime p^ which divides N, S must 
 transform some sub-group P^ of order jp/r into itself Since 
 q™ is not a factor of ^(P,-), some power of S must be per- 
 mutable with every operation of P,.. Hence, as in the theorem, 
 it follows that some power of S, certainly jS «"""', must be permu- 
 table with every operation of G. But no operation of the 
 group of isomorphism.s of G, except identity, is permutable 
 with every operation of G. Hence the group of isomorphisms 
 cannot contain an operation of order g™, if m > m' ; and there- 
 fore there is no isomorphism of G whose order contains a 
 
 higher power of q than g™' If then q^'q'^' , a number 
 
 relatively prime to N, is the order of an isomorphism of G, all 
 
 the numbers g'', q'''', are factors of S-(G), and so also is 
 
 their product. 
 
 178. The method, by which it has been shewn in § 175 that 
 
 the order of any isomorphism of P,. which transforms every 
 
 P,. 
 operation of each of the groups -p-^ and P^+i into itself is a 
 
 Pr+i 
 
179] OF A GIVEN GROUP 253 
 
 power of p, may be used to obtain the following more general 
 result. 
 
 Theorem X. If H is a self-conjugate sub-group of G, the 
 
 order of an isomorphism of G, which transforms every operation 
 
 C 
 of each of the groups jf ('''^d H into itself, is a factor of the order 
 
 ofH. 
 
 If 8 is any operation of G not contained in H, the 
 isomorphism will change 8 into 8h, where h is some operation 
 of H. If then m is the order of h, the isomorphism transforms 
 
 8,Sh,Sh\ ,Sh^-' 
 
 cyclically ; and therefore it transforms all the operations cif the 
 set 8H in cycles of m each. If S' is any operation of G not con- 
 tained in 8ff, the isomorphism will interchange the operations 
 of the set S'H among themselves in cycles of m' each, where m' 
 again is the order of some operation of H. The isomorphism, 
 when expressed as a substitution performed on the operations 
 
 of G, will consist of a number of cycles of m, m', symbols ; 
 
 and its order is therefore the least common multiple of 
 
 m, m', Now if q is any prime that divides the order 
 
 of H, and 5" the highest power of q that occurs as the order of 
 a cyclical operation of H, no power of q higher than g" can 
 
 occur in any of the numbers m, m', ; and q" is therefore 
 
 the highest power of q that can occur in their least common 
 multiple. This least common multiple, which is the order of 
 the isomorphism, must therefore divide the order of H. 
 
 179. Ex. 1. Shew that, for the group of order jo^ defined by 
 
 pp = l, QP = l, i?^=l, Q-'PQ = PR, 
 
 RP = PR, RQ = QR, 
 
 the symbol 
 
 P, Q, R 
 
 ( P, Q, ^\ 
 
 gives an isomorphism if 
 
 ^x- ay = n, (mod. p). 
 Hence determine the order of the group of isomorphisms. 
 
254 EXAMPLES [179 
 
 Ex. 2. Shew that, for the group of order p^ defined by 
 
 the symbol 
 
 P Q 
 
 ( P Q \ 
 Kp^Q", P'^fQ) 
 
 gives an isomorphism if x is not a multiple of p ; and determine the 
 order of the group of isomorphisms. 
 
 Ex. 3. Shew that the group of isomorphisms of the group of 
 order 2", defined by (§ 63) 
 
 F""-' =\,Q^ = P""', Q-^PQ = P~\ 
 
 is of order 2^""', when «->3. If m = 3, its order is 24 and it is 
 simply isomorphic with the last type but one of § 84. 
 
 Ex. 4. If is a complete group of order N, shew that the order 
 of K, the holomorph of G, is N^ ; and that the order of the holo- 
 morph of K is 2iV*. 
 
CHAPTER XII. 
 
 ON THE GRAPHICAL REPRESENTATION OF A GROUP'. 
 
 180. Our discussions hitherto have been confined ex- 
 clusively to groups of finite order. When however, as we now 
 propose to do, we consider a group in relation to the operations 
 that generate it, it becomes almost necessary to deal, incident- 
 ally at least, with groups whose order is not finite ; for it is not 
 possible to say a priori what must be the number and the nature 
 of the relations between the given generating operations, which 
 will ensure that the order of the resulting group is finite. 
 
 Many of the definitions given in respect of finite groups 
 may obviously be extended at once to groups containing an 
 infinite number of operations. Among these may be specially 
 mentioned the definitions of a sub-group, of conjugate opera- 
 tions and sub-groups, of self-conjugate sub-groups, of the 
 relation of isomorphism between two groups and of the 
 factor-group given by this relation. In regard to the last of 
 them, the isomorphism between two groups, one at least of 
 which is not of finite order, may be such that to one operation 
 of the one group there correspond an infinitely great number of 
 operations of the other. On the other hand, all the results 
 obtained for finite groups, which depend directly or indirectly on 
 the order of the group, necessarily become meaningless when 
 the group is not a group of finite order. 
 
 1 The investigations of this Chapter are due to Dyok, " Griippentheoretische 
 Studien," Math. Ann., Vol. xx, (1882), pp. 1—44. We have followed Dyck's 
 memoir closely except in two respects. Firstly, we have used a rather more 
 definite geometrical operation than that of the memoir ; and secondly, we have 
 not specially considered a regular and symmetric division of a closed surface, 
 apart from a merely regular division. 
 
256 GENEEAL AND SPECIAL [181 
 
 181. Suppose that 
 
 represent any n distinct operations which can be performed, 
 directly or inversely, on a common object, and that. between 
 these operations no relations exist. Then the totality of the 
 operations represented by 
 
 Sp'^S/8r^ , 
 
 where the number of factors is any whatever and the indices 
 are any positive or negative integers, form a group G of infinite 
 order, which is generated by the n operations. If, moreover, 
 whenever such a succession of factors as Sp'Sp^ occurs in the 
 above expression, it is replaced by »Sp''+^, each operation of 
 the group can be expressed in one way and in one way only by 
 an expression of the above form, which is then called reduced. 
 
 It will sometimes be convenient to avoid the use of negative 
 indices in the expression of any operation of the group. To this 
 end we may write 
 
 so that ^„+i is a definite operation of the group ; then 
 
 ,S',.~i = /Sr+,>SV+2 SnSn+iSi Sr-i, (^=1, 2, , n). 
 
 By using these relations to replace all negative powers of 
 operations wherever they occur, we may represent every 
 operation of the group in a single definite way by means of 
 the n + l operations 
 
 with positive indices only. 
 
 The group, thus defined and represented, is the most 
 general group conceivable that is generated by n distinct 
 operations. Any two such groups, for which n is the same, are 
 simply isomorphic with each other. 
 
 Suppose now that 
 
 represent n distinct operations, but that, instead of being entirely 
 independent, they are connected by a relation of the form 
 
 which will be represented by 
 
182] GROUPS 257 
 
 If is the group generated by these operations, an isomorph- 
 ism may be established between G and by taking Si 
 
 (i=l, 2, , n) as the operation of G that corresponds to 
 
 the operation Si of G. 
 
 Then to every operation of G 
 
 Sp'Sg^Sry 
 
 will correspond a single definite operation 
 
 .Sp-S/Sry. 
 
 of G; for the supposition that two distinct operations of G 
 correspond to the same operation of G leads to the result that 
 between the generating operations of G there is a relation, 
 which is not the case. On the other hand, to the identical 
 operation of G there will correspond an infinite number of 
 distinct operations of G, namely those which are formed by 
 combining together in every possible way all operations of G of 
 the form 
 
 R-V(Si)R, 
 
 where R is any operation of G. These operations of G form a 
 
 self-conjugate sub-group H; the corresponding factor-group 
 
 G . . . . . - 
 
 -p. is simply isomorphic with G. 
 
 If between the generating operations of G there are several 
 independent relations 
 
 MSi) = i,MSi)=i, ,MSi) = 'i-. 
 
 it may be shewn exactly as before that the groups G and G are 
 isomorphic in such a way that to the identical operation of G 
 there corresponds that self-conjugate sub-group of G, which is 
 formed by combining in every possible way all the operations 
 of G of the form 
 
 R-^fj{Si)R, 0' = 1. 2, ,m). 
 
 182. We may at once extend the result of the preceding 
 paragraph in the following way : — 
 
 Theorem I. If G is the group generated hy the n operations 
 between which the m relations 
 
 MSi) = i, MSi) = i, , MSi) = i, 
 
 B. 17 
 
258 GENERAL AND SPECIAL [182 
 
 exist ; and if is the group generated by the n operations 
 
 which are connected by the same m relations 
 
 as hold between the generating operations of 0, and by the 
 further m! relations 
 
 g^(Si) = l, g^(Si) = l, , g,n'(Si) = l; 
 
 — C 
 
 then is simply isomorphic with the factor-group ■=. ; where H 
 
 is that self-conjugate sub-grovp of G, which results from com- 
 bining in every possible way all operations of the form 
 
 R-'gj(Si)R, 0' = 1.2, ,m'), 
 
 R being any operation of G. 
 
 In proving this theorem, it is sufSScient to notice that, if we 
 
 take Si (i = 1,2, , m) as the operation of G which corresponds 
 
 to the operation Si of G, then to each operation of G' a single 
 definite operation of G will correspond, while to the identical 
 operation of G there corresponds the self-conjugate sub-group 
 HofG. 
 
 The theorem just stated is of such a general nature that it is 
 perhaps desirable to illustrate it by considering shortly some simple 
 examples. 
 
 Let us take first the case of a group G, generated by two in- 
 dependent operations *S'i and S^, subject to no relations ; and let us 
 suppose that the single relation 
 
 holds between the generating operations of G. The self-conjugate 
 sub-group H oi G then consists of all the operations 
 
 a an Q Pn 
 Oi O2 
 
 of G which reduce to identity if we regard S-^ and S^ as permutable ; 
 or, in other words, of those operations of G for which the relations 
 
 simultaneously hold. 
 
 In respect of this sub-group, the operations of G can be divided 
 into an infinite number of classes of the form 
 
182] GROUPS 259 
 
 For the operations of the class Sj^Sj^H, multiplied by those of 
 the class S/S^ H, give always operations of the class 
 
 since 5' is a self-conjugate sub-group ; and, because 
 
 while Sr^S^iS/S^'" belongs to H, the class S^" S^" S/ S/ 11 is the 
 same as Sf*^S^*'^H. Hence the operations of any two given classes, 
 multiplied in either order, give the same third class ; and therefore 
 
 G 
 
 the group -jf is an Abelian group generated by two permutable, 
 
 but otherwise unrestricted, operations. 
 
 As a second illustration, we will choose a case in which G is of 
 finite order. Let G be generated by the operations S and T, which 
 satisfy the relations 
 
 S^=l, y'=l, {STf=l; 
 
 and for G, suppose that the generating operations satisfy the add- 
 tional relation 
 
 Then H is formed by combining in all possible ways the 
 operations 
 
 Now it may be easily verified that in G, the operation ST^ 
 belongs to a set of three conjugate operations 
 
 ST\ TST, T^S; 
 
 and that these three operations are permutable among themselves, 
 while their product is identity. Hence H consists of the Abelian 
 group 
 
 and in respect of H, G may be divided into 27 classes of the form 
 
 ^0= (^rj,y (rp^rjy jj^ (^^ y, s = 0, 1, 2). 
 
 The group G will be defined by the laws according to which 
 these 27 classes combine among themselves; and the reader will 
 have no difficulty in verifying that it is isomorphic with the non- 
 Abelian group of order 27, whose operations are all of order 3 (§ 73). 
 
 Ex. If 5'i (= 1), j^j, , <S'jv are the operations of a group G 
 
 of finite order, prove that the totality of the operations of the form 
 Sp'^Sif^SgS^ generate a self-conjugate sub-group H; and that the 
 
 G 
 group -=.is an Abelian group. Shew also that, if H' is a self -conju- 
 re 
 gate sub-group of G and if -^ is Abelian, then H' contains H. 
 
 (Miller, Quarterly Journ. of Math., Vol. xxviii, (1896), p. 266.) 
 
 17—2 
 
260 
 
 GRAPHICAL REPRESENTATION 
 
 [183 
 
 183. For the further discussion of a group, as defined by its 
 generating operations and the relations between them, a suitable 
 graphical mode of representation becomes of the greatest assist- 
 ance. To this we shall now proceed. 
 
 In the simple case in which the group is generated by a 
 single unrestricted operation, such a representation may be 
 constructed as follows. Let G^ and C'_i be two circles which 
 touch each other ; C^ and (7_2 the inverses of C'_i in C^ and G^ 
 in 0_i ; Gg and G_s the inverses of C_2 in Oj and G^ in (7_i, and 
 so on. These circles (fig. 1) divide the plane in which they are 
 
 Fig. 1. 
 
 drawn into an infinite number of crescent-shaped spaces. Sup- 
 pose now that the space between Cj and 0_i is left white, and 
 the spaces between Cj and G^ and between C_i and G_2 (on either 
 side of this white space) are coloured black; the next pair on 
 either side left white, the next coloured black, and so on. Then 
 any white space may be transformed into any other (and any 
 black into any other) by an even number of inversions at the 
 circles 0_i and Oj ; and if S denote the operation consist- 
 ing of an inversion at (7_i followed by an inversion at C^, 
 the space between (7_i and Cj will be transformed into 
 another perfectly definite white space by the operation S", 
 while conversely the operation necessary to transform the space 
 between 0_i and Oj into any other given white space will be a 
 definite power of S. Hence if one of the white spaces, say 
 
183] 
 
 OF A CYCLICAL GROUP 
 
 261 
 
 that between G^^ and C^, is taken to correspond to the identical 
 operation, there is then a unique correspondence between the 
 white spaces and the operations of the group generated by 
 the unrestricted operation S ; and the figure that has been 
 constructed gives a graphical representation of the group. It 
 should be noticed that the actual geometrical process of in- 
 version, which has been here used to construct the spaces 
 corresponding to the operations of the group, is in no way 
 essential to the graphical representation. It is however con- 
 venient as giving definiteness to the construction; and later, 
 when we deal with the case of a general group, such definite- 
 ness becomes almost a necessity. 
 
 In a precisely similar manner, the group generated by a 
 single relation S, satisfying the relation 
 
 /S™ = 1, 
 may be treated. In this case, we take two circles C_i and 
 
 (7i intersecting at an angle — , and from these form, as before, 
 
 the circles obtained by successive inversions. This gives a 
 
 finite series of n circles, each of which intersects the two next to 
 it on either side at angles — , while the n circles divide the 
 
262 GRAPHICAL REPRESENTATION [183 
 
 plane into 2n spaces. If these are left white and coloured 
 black in alternate succession, and if one of the white is taken 
 to correspond to the identical operation, there is a unique 
 correspondence between the white spaces and the operations of 
 the group generated by S, where S represents the result of 
 successive inversions first at C'_i and then at C^. 
 
 This operation obviously satisfies the relation 
 
 and.;no simpler relation; so that the figure gives a graphical 
 representation of a cyclical group of order n. 
 
 The systems of circles in figures 1 and 2 have a common 
 geometrical property which may be noticed here as it will be of 
 use in the sequel. Successive inversions at any one of the pairs 
 C_i and (7i, Oj and G^, G^ and G^ are equivalent to the operation 
 S ; and therefore successive inversions at (7_i and 0, are equi- 
 valent to the operation /S*". Hence the result of an even 
 number of inversions at any of the circles in either figure is 
 equivalent to some operation of the group that the figure 
 represents. 
 
 184. We may now proceed to construct a graphical 
 representation of the group which is generated by n opera- 
 tions subject to no relations. To this end, suppose n-\-\ circles 
 drawn, each of which is external to all the others while each 
 touches two and only two of the rest. Such a system can be 
 drawn in an infinite variety of ways : we will suppose, to give 
 definiteness and simplicity to the resulting figure, that the 
 n -)- 1 points of contact lie on a circle, which cuts the n + 1 
 circles orthogonally. If these ?i, 4- 1 points taken in order are 
 
 Ai, J.2, , An+i, the successive circles are A^^^A^, A^A^, 
 
 , AnAn+i- -We will suppose that only so much of these 
 
 circles is drawn as lies within the common orthogonal circle 
 
 A1A2, An+i. The n + 1 circular arcs An+iA^, A1A2 
 
 then bound a finite simply connected plane figure which we 
 will denote provisionally by P. Suppose now that P is inverted 
 in each of its sides, that the resulting figures are inverted in 
 each of their new sides, and so on continually. Then from 
 their mode of formation no two of the figures thus arising can 
 
184] 
 
 OF A GENERAL GROUP 
 
 263 
 
 overlap either wholly or in part; and when the process is 
 continued without limit, every point in the interior of the 
 
 orthogonal circle A1A2 -^n+i will lie in one and only one of 
 
 the figures thus formed from P by successive inversions. 
 
 If P' is any one of the figures or polygons so formed, the set 
 of inversions by which it is derived from P is perfectly definite. 
 For suppose, if possible, that P' is derived from P by two 
 distinct sets of inversions represented by 2 and S'. Then 
 2S'~' is a set of inversions in the sides of P which transforms 
 P into itself But every set of inversions necessarily trans- 
 forms P into some polygon lying outside it, and therefore 
 
 22'-' = 1 ; 
 or the set of inversions composing 2 is identical term for term 
 
 with the set composing 2'. It immediately follows that the 
 polygons can be divided into two sets, according as they are 
 derived from P by an even or an odd number of inversions. The 
 
264 GRAPHICAL REPRESENTATION [184 
 
 latter we shall suppose coloured black, and the former (includ- 
 ing P) left white. Every white polygon will be surrounded 
 by black polygons and vice versS,. Since there is only one 
 definite set of inversions that will transform P into any other 
 white polygon P', the n+ 1 corners of P' will correspond one 
 by one to the n + 1 corners of P ; and when the perimeters of 
 the two polygons are described in the same direction of rotation 
 with regard to their interiors, the angular points that correspond 
 will occur in the same cyclical order. On the other hand, in 
 order that the corresponding angular points of a white and a 
 black polygon may occur in the same cyclical order, their peri- 
 meters must be described in opposite directions. In consequence 
 of these results, we may complete our figure (fig. 3) by lettering 
 every angular point of every polygon with the same letter that 
 occurs at the corresponding angular point of the polygon P. 
 
 185. If now Ti, Tj, , Tn+i represent inversions at 
 
 AjA^, A2A3, , An+iA-i, the operation Tr-i^r leaves the 
 
 comer Ay of P unchanged and it transforms P into the next 
 white polygon which has the corner ^4^ in common with P, the 
 direction of turning round Ar coinciding with the direction 
 
 An+iAn -4i of describing the perimeter of P. For brevity, 
 
 we shall describe this transformation of P as a positive rotation 
 round A^. If then we denote the operation IV-i^'r by the 
 single symbol Sr, we may say that Sr produces a positive 
 rotation of P round Ar. Let P, be the new polygon so 
 obtained; and let P/ be the polygon into which any other 
 white polygon P' is changed by a positive rotation round the 
 corner of P' that corresponds to Ay. Then if 2 is the set of 
 inversions that changes P into P', it also changes Pj into P/ : so 
 that "Z-^Brt changes P' into P/, i.e. produces a positive rotation 
 round the corner A,, of P' ; and /Sf^S changes P into P/. 
 
 Let us now represent the operations 
 
 m rp m rp T T 
 
 -'■n+l-'-i! -'■l-'-i! > -'■n-'-n+lt 
 
 by 'S'l, «'2> ) '^m+iJ 
 
 so that S182 SnSn+i = 1- 
 
 Then every operation, consisting of a pair of inversions in 
 
 the sides of P, can be represented in terms of 
 
 Oij O2) ^3> J ^n' 
 
185] OF A GENERAL GROUP 265 
 
 For an inversion at A^A^j^-^, followed by an inversion at A^Ag+i, 
 is given by TrTg ; and 
 
 ■'■r-'-s — J-r-'- r+1 • -^ r+i -^ r+2 • • • -^ s— i -i s 
 
 = Oy4.iOy_|_2 Og. 
 
 If 5 > r, this is of the form required. If s < r, the term 
 8n+i that then occurs may be replaced by 
 
 a -i.cf -1 .cf-i.cf-i 
 
 Hence finally, every operation consisting of an even number of 
 inversions in the sides of P can be expressed in terms of 
 
 'S'l, S„, Ss, , Sn ; 
 
 and with a restriction to positive indices, every such operation 
 can be expressed in terms of 
 
 Now it has been seen that no two operations, each consisting 
 of a set of inversions in the sides of P, can be identical unless 
 the component inversions are identical term for term. Hence 
 no two reduced operations of the form 
 
 f(Si) 
 are identical ; in other words, the n generating operations 
 
 "1; "2> ! "n 
 
 are subject to no relations. 
 
 If then we take the polygon P to correspond to the identical 
 operation of the group generated by 
 
 each white polygon may be taken as associated with the 
 operation which will transform P into it. The foregoing dis- 
 cussion makes it clear that in this way a unique correspondence 
 is established between the operations of G and the system of 
 white polygons ; or in other words, that the geometrical figure 
 gives a complete graphical representation of the group. 
 
 Moreover, since the operation 2~'/SrS is a positive rotation 
 round the corner A^ of the polygon 2 (calling now P the 
 polygon 1), a simple rule may be formulated for determining 
 by a mere inspection of the figure what operation of the group 
 any given white polygon corresponds to. 
 
266 GRAPHICAL RKPRESENTATION [185 
 
 This rule may be stated as follows. Let a continuous line 
 be drawn inside the orthogonal circle from a point in the white 
 polygon 1 to a point in any other white polygon, so that every 
 consecutive pair of white polygons through which the line 
 passes have a common corner, a positive rotation round which 
 leads from the first to the second of the pair. This is always 
 possible. Then if the common corners of each consecutive pair 
 of white polygons through which the line passes, starting from 
 
 1, are Ap, Ag, , A^, Ag, the final white polygon corresponds 
 
 to the operation 
 
 186. The graphical representation of a general group we have 
 thus arrived at is only one of an infinite number that could be con- 
 structed ; and we choose this in preference to others mainly because 
 the form of the figure and the relative positions of the successive poly- 
 gons are readily apprehended by the eye. As regards the mere estab- 
 lishment of such a representation we might, still using the process of 
 inversion for the purpose of forming a definite figure, have started 
 with w -f 1 circles each exterior to and having no point in common 
 with any of the others. Taking as the figure P the space external 
 to all the circles and inverting it continually in the circles, we should 
 form a series of black and white spaces of which the latter woidd 
 again give a complete picture of the group. It is however only 
 necessary to begin the construction of such a figure in order to 
 convince ourselves that it would not appeal to the eye in the same 
 way as the figure actually chosen. 
 
 Moreover, as in the representation of a cyclical group, the process 
 of inversion is in no way essential to the representation at which we 
 arrive. Any arbitrary construction, which would give us the series 
 of white and black polygons having, in the sense of the geometry of 
 position, the same relative configuration as our actual figure, would 
 serve our purpose equally well. 
 
 187. If 2 is the operation which transforms P into P', and 
 if Q is the black polygon which has the side ArA^+i in common 
 with P, then S transforms Q into the black polygon which has 
 in common with P' the side corresponding to ArA^+i. If then 
 we take Q to correspond to the identical operation, any black 
 polygon will correspond to the same operation as that white 
 
 * The reader wlio refers to Prof. Djck's memoir should notice that the 
 definition of the operation S^ above given is not exactly equivalent to that used 
 by Prof. Dyck. With his notation, the white polygon here considered would 
 correspond to the operation Sj,S^ S,S,. 
 
188] Of A SPECIAL GROUP 267 
 
 polygon with which it has the side AyA^+i in common. In this 
 way we may regard our figure as divided in a definite way into 
 double-polygons, each of which represents a single operation of 
 the group. 
 
 188. We have next to consider how, from the representation 
 of a general group whose n generating operations are subject 
 to no relations, we may obtain the representation of a special 
 group generated by n operations connected by a series of 
 relations 
 
 Fj(8i)=l, (j=l,2, ,m). 
 
 It has been seen (§181) that to the identical operation of the 
 special group there corresponds a self-conjugate sub-group H of 
 the general group ; or in other words, that the set of operations 
 S-ff of the general group give one and only one operation in the 
 special group. 
 
 Hence, to obtain from our figure for the general group one 
 that will apply to the special group, we must regard all the 
 double-polygons of the set SH as equivalent to. each other ; 
 and if from each such set of double-polygons we choose one as 
 a representative of the set, the totality of these representative 
 polygons will have a unique correspondence with the operations 
 of the special group. 
 
 We shall first shew that a set of representative double- 
 polygons can always be chosen so as to form a single simply 
 connected figure. Starting with the double-polygon, Pj, that 
 corresponds to the identical operation of the general group as 
 the one which shall correspond to the identical operation 
 of the special group, we take as a representative of some 
 other operation of the special group a double-polygon, P^, 
 which has a side in common with Pj. Next we take as 
 a representative of some third operation of the special group 
 a double-polygon which has a side in common with either 
 Pi or Pj; and we continue the choice of double-polygons 
 in this way until it can be carried no further. The set of 
 double-polygons thus arrived at of necessity forms a single 
 simply connected figure C, bounded by circular arcs ; and no 
 two of the double-polygons belonging to it correspond to the 
 same operation of the special group. Moreover, in G there is 
 
268 GRAPHICAL REPRESENTATION [188 
 
 one double-polygon corresponding to each operation of the 
 special group. To shew this, let C" be the figure formed by 
 combining with C every double-polygon which has a side in 
 common with C; and form C" from C, C" from C", and so on 
 as C has been formed from C. From the construction of C it 
 follows that every polygon in C is equivalent, in respect of the 
 special group, to some polygon in G. Similarly, every polygon 
 in C" is equivalent to some polygon in C and therefore to some 
 polygon in ; and so on. Hence finally, every polygon in the 
 complete figure of the general group is equivalent to some 
 polygon in 0, in respect of the special group ; and therefore, 
 since no two polygons of G are equivalent in respect of the 
 special group, the figure G is formed of a complete set of repre- 
 sentative double-polygons for the special group. 
 
 Suppose now that *Sf is a double-polygon outside C, with a 
 side Ar'A'r+i belonging to the boundary of G. Within C there 
 must be just one polygon, say ST, of the set SH. If this 
 polygon lay entirely inside C, so as to have no side on the 
 boundary of G, every polygon having a side in common with it 
 would belong to G. Now since S and ST are equivalent, every 
 polygon having a side in common with S is equivalent to some 
 polygon having a side in common with ST. Hence since G 
 contains no two equivalent polygons, ST must have a side on 
 the boundary of G; and if this side is A,."A",.+i, the operation T 
 of H transforms A/A'^+i into A/'A",-+i. Moreover, no operation 
 of H can transform A/A'r+i into another side of G; for if this 
 were possible, G would contain two polygons equivalent to S. 
 It is also clear that, regarded as sides of polygons within G, 
 Ar'A'r+i and ^/'J.",.+i belong to polygons of different colours. 
 Hence a correspondence in pairs of the sides of G is established : 
 to each portion A^'A'r+i of the boundary of G, which forms 
 a side of a white (or black) polygon of G, there corresponds 
 another definite portion J./'J.",.+i, forming a side of a black (or 
 white) polygon of G, such that a certain operation of H and its 
 inverse will change one into the other, while no other operation 
 of H will change either into any other portion of the boundary 
 ofC. 
 
 The system of double-polygons forming the figure G, and 
 the correspondence of the sides of G in pairs, will now give a 
 
190] OF A SPECIAL GROUP 269 
 
 complete graphical representation of the group. For the figure 
 has been formed so that there is a unique correspondence 
 between the white polygons of C and the operations of the 
 group, such that until we arrive at the boundary the previously 
 obtained rule will apply ; and when we arrive at a polygon on 
 the boundary, the correspondence of the sides in pairs enables 
 the process to be continued. 
 
 189. From the mode in which the figure G has been formed, 
 no two of the figures GH can have a polygon in common, when 
 for H is taken in turn each operation of the self-conjugate 
 sub-group H of the general group G; also the complete 
 set contains every double-polygon of our original figure. This 
 set of figures, or rather the division of the original figure into 
 this set, will then represent in a graphical form the self-con- 
 jugate sub-group H of G. Moreover, the operations which 
 transform corresponding pairs of sides of G into each other will, 
 when combined and repeated, clearly suffice to transform G into 
 any one of the figures GH and will therefore form a set of 
 generating operations of H. 
 
 190. A simple example, in which the process described in the 
 preceding paragraphs is actually carried out, will help to familiarize 
 the reader with the nature of the process and will also serve to 
 introduce a further modification of our figure. The example we 
 propose to consider is the special group with two generating opera- 
 tions which are connected by the relations 
 
 S^=\, 82^=1, 3182 = 8281. 
 As a first step, we will take account only of the relation 
 
 81O2 ^ OgOj, 
 
 and form for this special group the figure C. All operations 
 
 Oi O2 ) 
 
 for which 5a„ and 2/8^ have given values, are in the special group 
 identical. We may thus select from the figure for the general group 
 the set of polygons 
 
 ^/«S'/ (a,/? = -00 to +00) 
 as a set of representative polygons ; and a reference to the diagram ' 
 (fig. 4) makes it clear that this set of polygons forms a figure with 
 a single bounding curve. The black polygon which corresponds to 
 the operation 2 has here been chosen as that which has the side 
 AiA2 in common with the white polygon S. 
 
 ' In fig. 4 the orthogonal circle, which is not shewn, is taken to be a straight 
 line. 
 
270 
 
 EXAMPLE 
 
 [190 
 
 Each double-polygon, except those of the set (S'l™, contributes two 
 sides to the boundary of C, one belonging to a white polygon and 
 
 Fig. 4. 
 
 one to a black. The polygons, which border C and have sides in 
 
 common with S^S-,^, &v%S-,S^S-P and S^^S^S-,^; and these, regarded 
 
 as operations of the special group, are equivalent to S^S^^'^ and 
 
 S^S-^-'^. Hence the correspondence between the sides of C is such 
 that 
 
 (i) to the side A-^A^ of the white polygon S^S-^ corresponds the 
 side A^A^ of the black polygon S^S-^^~'^ ; 
 
 (ii) to the side A-^A^ of the black polygon S^S^^ corresponds the 
 side A^A^ of the white polygon S^^S/^^^ 
 
 When we now take account of the additional relations 
 
 the figure C is found to reduce to a set of nine double-polygons, 
 which is completely represented by fig. 5. 
 
 In addition to the correspondences between the sides of G to 
 which those just written simplify when the indices of S^ and S^ are 
 reduced (mod. 3), we have now also the correspondences, indicated 
 in the figure by curved lines with arrowheads, which result from the 
 new relations. Our figure may be further modified in such a way that 
 its form takes direct account of these four new correspondences. 
 Thus without in any way altering the configuration of the double- 
 polygons, from the point of view of geometry of position, we may 
 continuously deform the figure so that the pairs of corresponding 
 sides indicated by the curved arrowheads are brought to actual 
 coincidence. When this is done, the resulting figure will have the 
 form shewn in fig. 6. The correspondence in pairs of the sides of 
 the boundary is indicated in the figure by full and dotted lines. 
 
190] 
 
 c 
 
 
 
 CO 
 
 i^,^,,^ 
 
 r 
 
 V 
 
 
 ■<r 
 
 — -I T-M 
 
 
 
 = -_---^-. ^ (/) 
 
 ^ 
 
 
 n 
 
 t ;::.^. ^^4 
 
 1 
 
 
 -< 
 
 J 
 
 f( 
 
 
 ■< ) 
 
 ^ 
 
 ^i 
 
 > 
 
 CO 
 
 ■< 
 
 <n ^^ 
 
 k 
 
 i 
 
 -< 
 
 271 
 
 w 
 
272 
 
 GRAPHICAL REPRESENTATION 
 
 [190 
 
 The two unmarked portions A^A^A-^ correspond, as also do the two 
 similar portions marked with a full line, and the two marked with a 
 dotted line. 
 
 It will be noticed that, in this final form of the figure for the 
 special group, direct account is taken of the finite order of the 
 generating operations S^ and S^ and also of the operation S^S^- The 
 simplification of the figure that results by thus taking account 
 directly of the finite order of the generating operations, and the 
 greater ease with which the eye follows this simplified representa- 
 tion, are immediately obvious on a comparison of figs. 5 and 6. 
 
 191. In the applications of this graphical representation 
 of a group that we have specially in view, namely to groups 
 of finite order, the generating operations themselves are 
 necessarily of finite order. The generating operations 
 
 of such a group may be taken as of orders 
 wij, m^, m„; 
 
191] OF A GROUP OF FINITE ORDER 273 
 
 and if SiS^ SnSn+i = l, 
 
 then Sn+i will be of finite order nin+i. We shall therefore next 
 consider a group generated by n operations which satisfy the 
 relations 
 
 sr^ = i, sr^^i, , -s„»«=i, (8,8, ,sf„r«+>=i. 
 
 The simple example we have given makes it clear that, 
 at least in some particular cases, relations of this form may 
 be directly taken into account in constructing our figure ; in 
 such a way that in the complete figure, consisting of a finite or 
 an infinite number of double-polygons, the correspondence in 
 pairs of the sides of the boundary, if any, will depend upon 
 further relations between the generating operations. 
 
 We may, in fact, always take account of relations of the 
 form in question in the construction of our figure as follows. 
 
 Let us take as before n+ 1 arcs of circles 
 
 bounding a polygonal figure P of n+1 corners ; but now, 
 instead of supposing the circles Ar-iA^ and ArA^+i to touch at 
 .4r, let them cut at an angle (measured inside P) of 
 
 mr 
 
 (r = l,2, ,n), 
 
 while AnAn+i and Ar^iA, cut at an angle . Such a figure 
 
 can again be chosen in an infinite variety of ways : we will 
 suppose that it is drawn so that the n + 1 circles have a common 
 orthogonal circle. This clearly is always possible ; but it is not 
 now necessarily the case that this orthogonal circle is real. Let 
 the figure P be now inverted in each of its sides ; let the new 
 figures so fornied be inverted in each of their new sides ; and so 
 on continually. Then since the angles of P are sub-multiples of 
 two right angles, no two of the figures thus formed can overlap 
 in part without coinciding entirely. Moreover, when the process 
 is completely carried out, every point within the orthogonal 
 circle when it is real, and every point in the plane of the figure 
 when the orthogonal circle is evanescent or imaginary, will lie 
 in one and in only one of the polygons thus formed from P by 
 successive inversions. 
 
 B. 18 
 
274 GRAPHICAL REPRESENTATION [192 
 
 192. Exactly as with the general group, these polygons 
 are coloured white or black according as they are derivable 
 from P by an even or an odd number of inversions. The 
 corners of any white polygon correspond one by one to the 
 corners of P ; so that, when the perimeters of the polygons are 
 described in the same direction, corresponding corners occur in 
 the same cyclical order. 
 
 If now the operation of successive inversions at An-^-^A^ and 
 J.iJ.2 is represented by S^, and that of successive inversions at 
 
 Ar-iAr and A^Ar+i by Sr,{r = 2, 3, , n); all operations, 
 
 consisting of an even number of inversions in the sides of P, 
 can be represented in terms of 
 
 Moreover, from the construction of the polygon P, these 
 operations satisfy the relations 
 
 sr' = i, s,^ = i, , >s„™« = i, s„+r"+^ = i, 
 
 where S1S2 SnSn+i = 1. 
 
 Again, if P' is any white polygon of the figure, which can be 
 derived from P by the operation 2, a positive rotation (§ 185) of 
 P' round its comer J./ is effected by the operation %~^SrX; 
 and, if P" is the polygon so obtained, P" is derived from P by 
 the operation S^'Z. It is to be observed that a positive rotation 
 of a polygon round its Ar corner is now an operation of finite 
 order wv. 
 
 Suppose now that two operations S and S' transform P into 
 the same polygon P', so that 22'"' leaves P unchanged. If 
 this operation, written at length, is 
 
 >Sf/ s/SryS/, 
 
 and if P is transformed into Pi by a positive rotation round 
 As repeated B times, Pj into Pj by a rotation round its 
 corner A^ repeated 7 times, and so on ; then the operation 
 may be indicated by a broken line drawn from P to Pj, from 
 Pi to P2, and so on, the line returning at last to P. But the 
 operation indicated by such a line is clearly equivalent to com- 
 plete rotations, (i.e. rotations each of which lead to identity). 
 
193] OF A GROUP OF FINITE ORDER 275 
 
 round each of the corners which the broken line includes. In 
 other words, 2S'~^ reduces to identity when account is taken of 
 the relations which the generating operations satisfy. Hence 
 finally, to every white polygon P' will correspond one and only 
 one of the operations of the group, namely that operation 
 which transforms P into P'. The same is clearly true of the 
 black polygons ; and by taking P and a chosen black polygon 
 which has a side in common with P as corresponding to the 
 identical operation, the required unique correspondence is 
 established between the complete set of double-polygons in the 
 figure and the operations of the group, the relations which the 
 generating operations satisfy being directly indicated by the 
 configuration of the figure. Moreover, as with the general 
 group (§ 185), a simple rule may be stated for determining, 
 from an inspection of the figure, the polygon that corresponds 
 to any given operation of the group. 
 
 193. The number of polygons in the figure and therefore 
 the order of the group will still, in general, be infinite. We 
 may now proceed, just as in the previous case of a quite general 
 group, to derive from the figure representing the group 0, 
 generated by n operations satisfying the relations 
 
 Sr^ = 1, S,^' = \, , S„™« = 1, , (5fi5f, ,§„)"'"+• = 1 , 
 
 a suitable representation of the more special group G, generated 
 by n operations which satisfy the above relations and in addition 
 the further m relations 
 
 MSi) = l, (j = l,2, ,m). 
 
 As has been seen in § 182, if H is the self-conjugate sub- 
 group of which is formed by combining all possible operations 
 of the form 
 
 and if 2 is any operation of G, then the set of operations IZH, 
 regarded as operations of G, are all equivalent to each other. 
 From each set of polygons XH in the figure of G, we may 
 therefore choose one to represent the corresponding operation 
 of G ; and, as was shewn with the general group, a complete 
 set of such representative polygons may be selected to form a 
 
 18—2 
 
276 GRAPHICAL REPRESENTATION [193 
 
 connected figure, i.e. a figure which does not consist of two or 
 more portions which are either isolated or connected only by 
 corners. Moreover, as in the former case, the sides of this 
 figure will be connected in pairs ^/J.',.+i and J./'.4'V+i, which 
 are transformed into each other by some operation T oi H and 
 its inverse, while no other operation of H will transform either 
 Ar'A'r+i or Ar"A"r+i into any other side of G. 
 
 It is not now however necessarily the case that the figure C, 
 as thus constructed, is simply connected. Let us suppose then 
 that C has one or more inner boundaries as well as an cuter 
 boundary, and denote one of these inner boundaries by L. If 
 the sides of L do not all correspond in pairs, and if Af'A'r+i is a 
 side of L such that the other side ^/'j1",.+i corresponding to it 
 does not belong to L, we may replace the double-polygon P" in 
 C of which Ar"A"r+i is a side by the double-polygon, not 
 previously belonging to C, of which J/^V+i is a side. If P" 
 has a side on the boundary L, the new figure C thus obtained 
 will have one inner boundary less than C; and if P" has no side 
 on the boundary L, the new inner boundary L' that is thus 
 formed from L will contain one double-polygon less than L, 
 while the number of inner boundaries is not increased. This 
 process may be continued till the new inner boundary L^ which 
 replaces L is such that all of its sides correspond in pairs. 
 
 Let now AgAg+i and A/A'g+i be a pair of corresponding 
 sides of ii, such that AgAsj^.^ is transformed into Ag'A's+i by an 
 operation h of the self-conjugate sub-group H. A side ^j^(+i 
 of another boundary of C may be chosen such that AgAg^^ 
 and AtAt+i are sides of a simply connected portion, say B,oiG; 
 while no side of L^ except AgAgj^^ forms part of the boundary of 
 B. The polygons of B are equivalent, in respect of the special 
 group, to those of Bh. Moreover, since the sides of L^ corre- 
 spond in pairs, no side of Bh, except AgA'g^^ can coincide with 
 a side of Zj. Hence when B is replaced by Bh, the inner 
 boundary L^ will be got rid of and no new inner boundary will 
 be formed. Finally then, G may always be chosen so as to form 
 a single simply connected figure. 
 
 The simply connected plane figure G, which has thus been 
 constructed, with the correspondence of the sides of its boundary 
 
194] OF A GROUP OF FINITE ORDER 277 
 
 in pairs, will now give a complete graphical representation of 
 the special group. The rule already formulated will determine 
 the operation of the group to which each white polygon 
 corresponds; and when, in carrying out this rule, we come to 
 a polygon on the boundary, the correspondence of the sides of 
 the boundary in pairs will enable the process to be continued. 
 
 The correspondence of the sides of C in pairs involves a 
 correspondence of the corners in sets of two or more. Thus if 
 Ar is a corner of G and if, of the m,. white polygons which in 
 the complete figure have a corner at Ar, Mi lie within G, there, 
 must within G be m^ — Wi white polygons equivalent to the 
 remainder, and each of these must have an A^ corner on the 
 boundary. If J.,.' is a corner of G such that there are n^ white 
 polygons, lying within G and having a corner at A^, and if one 
 of the sides of the boundary with a corner at A^ corresponds 
 to one of the sides of the boundary with a corner at A^, these 
 Wj white polygons must be equivalent to n.^ of the white polygons, 
 lying outside G and having a corner at J.^. If 
 
 til + W2 < l^r, 
 
 there must be a third corner Ar', contributing n^ more white 
 polygons towards the set. With this we proceed as before ; and 
 the process may be continued till the whole of the m^ white 
 polygons surrounding A^ are accounted for. The set of corners 
 
 A^, Ar, Ar" , will then form a set of corresponding corners, 
 
 which are equivalent to each other in respect of the special 
 group ; and the whole of the corners of G may be divided into 
 such sets. At each set of corresponding corners Ar of G there 
 must clearly be also m,. black polygons belonging to G ; and 
 the sum of the angles of at a set of corresponding comers 
 must be equal to four right angles. 
 
 194. When the order of the group is finite, we may still 
 further so modify our figure as to take account of the corre- 
 spondence of the sides of the boundary in pairs. We may, in 
 fact, by a suitable bending and stretching of the figure, bring 
 corresponding sides of the boundary to actual coincidence. 
 When this is done, the figure will no longer be a piece of a 
 plane with a single boundary, but will form a continuous 
 
278 
 
 GRAPHICAL REPRESENTATION 
 
 [194 
 
 surface, which is unbounded and in general is multiply con- 
 nected. Every point Ar on the surface, which in the plane 
 figure did not lie on the boundary, will be a corner common 
 to 2m^ polygons alternately black and white ; and, in con- 
 sequence of what has just been seen in regard to the corre- 
 spondence of corners of the boundary, the same is true 
 for every point Ar on the surface which in the plane figure 
 consisted of a set of corresponding corners of the boundary. If 
 N is the order of the group, the continuous unbounded surface 
 will be divided into 2iV polygons, black and white. The con- 
 figuration of the set of white polygons with respect to any one 
 of them will, from the point of view of geometry of position, be 
 the same as that with respect to any other; and the like is 
 true for the black polygons. Such a division of a continuous 
 unbounded surface is described as a regular division ; and we 
 have finally, as a graphical representation of any group of finite 
 order N, a division of a continuous surface into 2N polygons, 
 half black and half white, which is regular with respect to each 
 
 Fig. 7. 
 
195] OF A GROUP OF FINITE ORDER 279 
 
 set. The correspondence between the operations of the group 
 and the white polygons on the surface is given by the rule that 
 a single positive rotation of the white polygon X round its 
 comer Ay leads to the white polygon S,.S. 
 
 195. We may again here illustrate this final modification of the 
 graphical representation of a finite group by a simple example. For 
 this purpose, we choose the group defined by 
 
 S,'=l, S,'=l, S,'=l, 
 
 This group (§ 74) is a non-Abelian group of order 8, containing 
 a single operation of order 2. The reader will have no difiiculty in 
 verifying that the plane figure for this group is given by fig. 7 ; and 
 that opposite sides of the octagonal boundary correspond. The 
 single operation of order 2 is 
 
 this corresponds to a displacement of the triangles among themselves 
 in which all the six corners remain fixed. If now corresponding 
 sides of the boundary are brought to coincidence, the continuous 
 surface formed will be a double-holed anchor-ring, or sphere with 
 
280 GENUS [195 
 
 two holes through it. A view of one half of the surface divided 
 into black and white triangles, is given in fig. 8. The half of the 
 surface, not shewn, is divided up in a similar manner; and the 
 operation of order 2 replaces each triangle of one half by the 
 corresponding triangle of the other, an operation which clearly leaves 
 the six corners of the polygons undisplaced. 
 
 196. The form of the plane figure G, which with the corre- 
 spondence of its bounding sides in pairs represents the group, 
 is capable of indefinite modification by replacing individual 
 polygons on the boundary by equivalent polygons. If however 
 we reckon a pair of- corresponding sides of the boundary as a 
 single side and a set of corresponding corners of the boundary 
 as a single corner, it is clear that, however the figure may be 
 modified, the numbers of its corners, sides and polygons remain 
 each constant. This may be immediately verified on replacing 
 any single boundary polygon by its equivalent. 
 
 If now A be the number of corners, and E the number of 
 sides in the figure G when reckoned as above, 2N being the 
 number of polygons, then the connectivity^ 2^ + 1 of the closed 
 surface is given by the equation 
 
 2^ = 2-l-^-2iV-^. 
 
 When the group and its generating operations are given, 
 the integer p is independent of the form of the plane figure G, 
 which as has been seen is capable of considerable modification. 
 The plane figure G however depends directly on the set of 
 generating operations that is chosen for the group. For a given 
 gi'oup of finite order, such a set is not in general unique ; and 
 the number of generating operations as well as their order will 
 in general vary from one set to another. It does not necessarily 
 follow, and in fact it is not generally the case, that the con- 
 nectivity of the surface by whose regular division the group is 
 represented, is independent of the choice of generating opera- 
 tions. There must however obviously be a lower limit to the 
 number p for any given group of finite order, whatever 
 generating operations are chosen ; this we shall call the genus 
 of the group ^ 
 
 ' Forsyth, Theory of Functions, p. 325. 
 
 ' Hurwitz, "Algebraisclie Gebilde mit eindeutigen Transformationen in 
 aich," Math. Ann. xli, (1893), p. 426. 
 
197] OF A GROUP 281 
 
 197. We shall now shew that there is a limit to the order 
 of a group which can be represented by the regular division of 
 a surface of given connectivity 2p+ 1. If iV is the order of 
 such a group, generated by the n operations 
 
 which satisfy the relations 
 
 Sr^ = l, 8r' = l, , -sv—i, 
 
 >Sii*Si2 Sn = 1 ; 
 
 the surface will be divided in 2iV polygons of n sides each. 
 Let Ai, A2, ....... An be the angular points of one of these 
 
 polygons ; and suppose that on the surface there are G^ corners 
 in the set to which A^ belongs, G^ in the set to which A^ belongs, 
 and so on. Round each corner J.,, there are 2m^ polygons ; and 
 each polygon has one and only one corner of the set to which 
 Ar belongs. Hence 
 
 Grinir = N, 
 
 n I 
 
 andso G-^ + G^^ + C'„ = iVS— . 
 
 1 rrir 
 
 Again, each side belongs to two and only to two polygons, so 
 that the number of sides is 
 
 Nn. 
 
 Using these values for A and E in the formula of § 196, 
 we obtain the equation 
 
 2(^-l) = i\r(._2-U 
 
 A complete discussion of this equation for the cases p = Q 
 and p = \ will be given in the next chapter. 
 
 When ^ is a given integer greater than unity, we can 
 determine the greatest value that is possible for N by finding 
 the least possible positive value of the expression 
 
 n 1 
 
 n-2-t—. 
 1 rrir 
 
 If n > 4, this quantity is not less than ^, since ?;i,. cannot be 
 less than 2. 
 
 If w = 4, the simultaneous values 
 
 mi = ma = wis = wii = 2 
 
282 LIMITATION ON THE ORDER [197 
 
 are not admissible, since they make the expression zero. Its 
 least value in this case will therefore be given by 
 
 nil = m2 = ms = 2, mi = 3 ; 
 
 and the expression is then equal to ^. 
 
 If re = 3, we require the least positive value of 
 
 «ii nia mj 
 
 Now the three sets of values 
 
 nil = 3, mj = 3, mg = 3, 
 
 mi = 2, ma = 4, mj = 4, 
 
 and mi = 2, m^ = S, m, = 6, 
 
 each make iT zero ; and therefore no positive value of K can be 
 less than the least of those given by 
 
 mi = 3, m., = 3, ms = 4, 
 
 m^ = 2, wij = 4, ma = 5, 
 
 and mi = 2, m^ = 3, m.i = l. 
 
 These sets of values give for K the values j*^, JL and J^. 
 Hence finally, the absolutely least positive value of the expression 
 is Jj, and therefore the greatest admissible value of N is 
 
 84 {p - 1). 
 Hence' : — 
 
 Theorem II. The order of a group, that can he represented 
 by the regular division of a surface of connectivity 2p + 1, cannot 
 exceed 84 ( j3 — 1 ), ^^ heing greater than unity. 
 
 198. If, when a group is represented by the regular division 
 of an unbounded surface, we draw a line from any point inside 
 the white polygon 1 (or any other polygon) returning after any 
 path on the surface to the point from which it started, it will 
 represent a relation between the generating operations of the 
 group. For in following out along the line so drawn the rule 
 that determines the operation of the group corresponding to 
 each white polygon, some operation 
 
 ' Hurwitz, loc. cit. p. 424. 
 
198] OF A GROUP OF GIVEN GENUS 283 
 
 will be found to correspond to the final polygon ; and this 
 being the white polygon 1, it follows that 
 
 If the surface is simply connected, any such line can be 
 continuously altered till it shrinks to a point ; and therefore the 
 n + 1 relations between the n generating operations completely 
 define the group, since all other relations can be deduced from 
 them. 
 
 If however the surface is of connectivity 2p + l, there are 
 2p independent closed paths that can be drawn on the surface, 
 no one of which can by continuous displacement either be 
 shrunk up to a point or brought to coincidence with another ; 
 and every closed path on the surface can by continuous dis- 
 placement either be brought to a point or to coincidence with 
 a path constructed by combination and repetition of the 2p in- 
 dependent paths'. Any one of these 2p independent paths will 
 give a relation between the n generating operations of the 
 group, which cannot be deduced from the ra + 1 relations on 
 which the angles of the polygons depend. Moreover, every rela- 
 tion between the geaerating operations can be represented by a 
 closed path on the surface ; and therefore there can be no further 
 relation independent of the original relations and those obtained 
 from the 2p independent paths. There cannot therefore be 
 more than 2p independent relations between the n generating 
 operations of a group, in addition to the n + l relations that give 
 the order of the generating operations and of their product ; 
 2^-1-1 being the connectivity of the surface by whose regular 
 division into w-sided polygons the group is represented. 
 
 The 2p relations given by 2p independent paths on the 
 surface are not, however, necessarily independent. In fact we 
 have already had an example to the contrary in § 195. On the 
 closed surface, by the regular division of which the group there 
 considered is represented, four independent closed paths can be 
 drawn. Any three of the corresponding relations can be derived 
 from the fourth by transformation. 
 
 The only known cases in which the 2p relations are in- 
 dependent are those of a class of groups of genus one (§ 205). 
 ' Forsyth, Theory of Functions, p. 330. 
 
284 NOTE TO § 194 [198 
 
 Ex. Draw the figure of the group generated by .S"] , S^, S^, where 
 
 s,'=i, s,' = \, s,<' = i, s,SA = i. 
 
 Shew from the figure that the special group, given by the 
 additional relation 
 
 is a finite group of order 48 ; and that it can be represented by the 
 regular division of a surface of connectivity 5. 
 
 Note to § 194. 
 
 If in the process of bending and stretching, described in § 194, 
 by means of which the plane figure C is changed into an unbounded 
 surface, the angles of the polygons all remain unaltered, the circles 
 of the plane figure will become continuous curves on the surface. 
 These curves on the surface, which we will still call circles, are 
 necessarily re-entrant. It is not however necessarily the case that, 
 on the surface, a circle will not cut itself. 
 
 In the plane figure for the general group, an inversion at any 
 circle of the figure leaves the figure unchanged geometrically but 
 interchanges the black and white polygons. Each circle is, in fact, a 
 line of symmetry for the figure such that, in respect of it, there 
 is corresponding to every white polygon a symmetric black polygon 
 and vice versa. 
 
 Similarly on the surface a circle which does not cut itself may 
 be a line of symmetry, such that a reflection at it is an operation of 
 order two which leaves the surface and its division into polygons 
 unchanged, but interchanges black and white polygons. When this 
 is the case, every circle on the surface will be a line of symmetry and 
 no circle will cut itself. On the other hand no such operation can 
 ever be connected with a circle which cuts itself. 
 
 When such lines of symmetry exist. Prof. Dyck speaks of the 
 division of the surface as regular and symmetric. 
 
CHAPTER XIII. 
 
 ON THE GRAPHICAL REPRESENTATION OF GROUPS : 
 GROUPS OF GENUS ZERO AND UNITY : CAYLEY's 
 COLOUR GROUPS. 
 
 199. We shall now proceed to a discussion in the cases 
 p = and jp = 1 of the relation 
 
 2(^-l) = i^(n-2-|i-), 
 
 which connects the number and the orders of the generating 
 operations of a group with the order of the group itself; and to 
 the consideration of the corresponding groups. 
 
 For any given value of p, other than p = l, we may regard 
 this relation as an equation connecting the positive integers 
 
 N,n,ini,m2, , m„. It does not however follow from the 
 
 investigations of the last Chapter that there is always a 
 group or a set of groups corresponding to a given solution of 
 the equation. In fact, for values oi p greater than 1, this is not 
 necessarily the case. We shall however find that, when ^ = 0, 
 there is a single type of group corresponding to each solution 
 of the equation; and that, when p = l, there is an infinite 
 number of types of group, all characterized by a common 
 property, corresponding to each solution of the equation. 
 When p = 0, the groups are (§ 196) of genus zero; and all 
 possible groups of genus zero are found by putting ^ = 
 
286 GROUPS [199 
 
 in the equation. The groups thus obtained are of special 
 importance in many applications of group-theory ; for this 
 reason, they will be dealt with in considerable detail. 
 
 200. When p = 0, the equation may be written in the form 
 
 in this form, it is clear that the only admissible values of n are 
 2 and 3. 
 
 First, let n = 2. The only possible solution then is 
 
 N = lUi = m^ = n, 
 
 n being any integer. The corresponding group is a cyclical 
 group of order n. 
 
 Secondly, ]et n—S. In this case, one at least of the three 
 integers wii, mg, ms must be equal to 2, as otherwise the right- 
 hand side of the equation would be not less than 2. We may 
 therefore without loss of generality put mj = 2. If now both 
 m^ and wis were greater than 3, the right-hand side would still 
 be not less than 2 ; and therefore we may take m^ to be either 
 2 or 3. When mj and m^ are both 2, the equation becomes 
 
 giving m3 = n, N= 2n, 
 
 where n is any integer. 
 
 When wii is 2 and mj is 3, the equation is 
 2_ 1_ J^ 
 
 This has three solutions in positive integers ; namely, 
 m, = S, iV"=12; 
 m3 = 4, iV"=24; 
 and m-j = 5, iV= 60. 
 
202] OF GENUS ZERO 287 
 
 The solutions of the equation for the case p = may there- 
 fore be tabulated in the form : — 
 
 
 TOj 
 
 ^2 
 
 ™3 
 
 N 
 
 I 
 
 n. 
 
 It 
 
 
 n 
 
 II 
 
 2 
 
 2 
 
 m 
 
 2b 
 
 III 
 
 2 
 
 3 
 
 3 
 
 12 
 
 IV 
 
 2 
 
 3 
 
 4 
 
 24 
 
 V 
 
 2 
 
 3 
 
 5 
 
 60 
 
 201. That a single type of group actually exists, corre- 
 sponding to each of these solutions, may be seen at once by 
 returning to our plane figure. The sum of the internal angles 
 of the triangle J.1.42J.3 formed by circular arcs is, in each of 
 these cases, greater than two right angles ; and the common 
 orthogonal circle is therefore imaginary. The complete figure 
 will therefore divide the whole plane into black and white 
 triangles, so that there are no boundaries to consider. More- 
 over, the number of white triangles in each case must be equal 
 to the corresponding value of iV; for the preceding investigation 
 shews that this is a possible value, and on the other hand the 
 process, by which the figure is completed from a given original 
 triangle, is a unique one. There is therefore a group corre- 
 sponding to each solution; and the correspondence which has 
 been established in any case between the operations of a group 
 and the polygons of a figure, proves that there cannot be two 
 distinct types of group corresponding to the same solution. 
 
 202. The plane figure for p=0 does not, in fact, differ 
 essentially from the figure drawn on a continuous simply con- 
 nected surface in space. The former may be regarded as the 
 stereographic projection of the latter. The five distinct types 
 are represented graphically by the following figures. 
 
 The first is a cyclical group, and the figure (fig. 9) does not 
 differ essentially from fig. 2 in § 183. 
 
 The group given by the second solution of the equation 
 is called the dihedral group. It is represented by fig. 10. 
 
288 
 
 GROUPS 
 
 [202 
 
 Fig. 9. 
 
 Fig. 10. 
 
202] 
 
 OF GENUS ZERO 
 
 289 
 
 Fig. 11. 
 
 The group given b};- the third solution of the equation is 
 represented in fig. 11. It is known as the tetrahedral grotxp. 
 
 To the fourth solution of the equation corresponds the group 
 represented by fig. 12. It is known as the octohedral group. 
 
 To the fifth solution of the equation corresponds the group 
 represented in fig. 13. It is known as the icosahedral group. 
 
 The four last groups are identical with the groups of rotations 
 which will bring respectively a double pyramid on an m-sided 
 base, a tetrahedron, an octohedron, and an icosahedron to co- 
 incidence with itself in each case^ 
 
 Whea the figures are drawn on a sphere, and the three circles 
 of the original triangle and therefore also all the circles of the 
 figure are taken to be great-circles of the sphere, the actual dis- 
 placements of the triangles among themselves which correspond 
 
 ' Klein, " Vorlesungen iiber das Ikosaeder," Chap. i. 
 
 19 
 
Fig. 12. 
 
 Fig. 13. 
 
203] 
 
 GROUPS OF GENUS ZERO 
 
 291 
 
 to the operations of the group can be effected by real rotations 
 about diameters of the sphere ; thus the statement of the pre- 
 ceding sentence may be directly veriiied. 
 
 203. In terms of their generating operations, the^ve types 
 of group of genus zero are given by the relations : — 
 
 I. 
 
 ,Si» = l, 
 
 8,^ = 1, 
 
 8A=1 
 
 II. 
 
 'Si' = l, 
 
 8,' = I, 
 
 S,- = l, 8A8, = 1 
 
 III. 
 
 S,'=l, 
 
 S,' = l, 
 
 8,^=1, 8,8 A=l 
 
 IV. 
 
 S^' = l, 
 
 8,' = 1, 
 
 8,^ = 1, S,8A=^1 
 
 V. 
 
 8^' = !, 
 
 S,' = l, 
 
 S,' = l, S.SA^l. 
 
 The first of these does not require special discussion. 
 In the dihedral group, we have 
 
 818381 = 8182 = 8f'^. 
 
 The dihedral group of order 2?i therefore contains a cyclical 
 sub-group of order n self-conjugately ; and every operation of 
 the group which does not belong to this self-conjugate sub- 
 group is of order 2. The operations of the group are given, 
 each once and once only, by the form 
 
 Si-S/, (a = 0, 1; ^ = 0, 1, 2, ,n-l). 
 
 When n = 3, this group is simply isomorphic with the 
 symmetric group of three symbols. 
 In the tetrahedral group, since 
 
 (SAy = i, 
 
 82~^SiS28i = S28i8f^, 
 
 and therefore S,, S2~'^8iS2, (SaSj/Sa"' are permutable with each 
 other. These operations of order 2 (with identity) form a self- 
 conjugate sub-group of order 4 ; and the 12 operations of the 
 group are therefore given by the form 
 
 S2'^'(82-'8Ay8iy, 
 or 82'^S/82Siy, (a = 0, 1, 2 ; ;8, 7 = 0, 1). 
 
 If ,Sf, = (12)(34), >S, = (123), 
 
 then 8182 = (IM): 
 
 and therefore the tetrahedral group is simply isomorphic with 
 the alternating group of four symbols. 
 
 19—2 
 
292 GROUPS OF GENUS ZERO [203 
 
 If, in the octohedral group, we write 
 
 then S^S' = S^S, = SA^S, ; 
 
 and therefore (S^SJ = 1. 
 
 Hence 8^ and S' generate a tetrahedral group. 
 
 Again S,SA = {S,S'y, 
 
 and 8,8' S, =-- 8,8'8f-\ 
 
 so that this is a self-conjugate sub-group. The operations of 
 the group are given, each once and once only, in the form 
 S,-8/S,'y8A'^, (a, -y, S = 0, 1 ; /3 = 0, 1, 2). 
 
 If 8, = (12), S,= {234>), 
 
 then 8,8^ = (134,2) ■ 
 
 and therefore the octohedral group is simply isomorphic with 
 the symmetric group of four symbols. 
 
 The icosahedral group is simple. It is, in fact, simply 
 isomorphic with the alternating group of five symbols which 
 has been shewn (§ 111) to be a simple group. Thus if 
 
 /Sf, = (12)(34), -S3 =(135), 
 then ,81-82 = (1234.5); 
 
 so that the substitutions 8, and 8^ satisfy the relations 
 
 8^ = 1, -83^ = 1, (8Ay = i. 
 
 They must therefore generate an icosahedral group or one 
 of its sub-groups. On the other hand, from the substitutions 
 Si and -82 all the even substitutions of five symbols may be 
 formed, and these are 60 in number. The group therefore 
 cannot be a sub-group of the icosahedral group; the only 
 alternative is that the two are identical. 
 
 As the icosahedral group has no self- conjugate group, we 
 cannot in this case so easily construct a form which will 
 represent each operation of the group just once in terms of 
 the generating operations. It is however not difficult to verify 
 that this is true of the set of forms^ 
 
 8.^8A^8A^8„ («,^- 0,1, 2, 3. 4). 
 
 1 Dyck, " Gruppentheoretische Studien," Math. Ann. xx, (1882), p. 35, and 
 Klein, loc. cit. p. 26. 
 
205] GROUPS OF GENUS ONE 293 
 
 204. We shall next deal with the equation in the case 
 p = l. In this case alone, the order of the group disappears 
 from the equation, which merely gives a relation between the 
 number and order of the generating operations. This may be 
 written in the form 
 
 2 = 2(1-^); 
 1 V mj 
 
 and n must therefore be either 4 or 3. 
 When n is 4, the equation becomes 
 
 „ 1111 
 
 2 =— + — + — + — , 
 mi ma mj m^ 
 
 and the only solution is clearly 
 
 rrii = ma = mg = mi = 2. 
 
 When n is 3, the equation takes the form 
 
 111 
 
 1=—+-+—, 
 Ml ma ms 
 
 and is easily seen to have three solutions, viz. 
 
 mi = 8, ma = 3, mg = 3 ; 
 
 mi = 2, ma = 4, ms = 4 ; 
 and mi = 2, wia = 3, m^ = 6. 
 
 205. Take first the solution 
 
 ?i = 4, mi = m2 = m3 = m4= 1. 
 The corresponding general group is defined by the relations 
 
 S,^=l, S,'=l, S,^ = l, »S'/=1, 
 8182838^= 1. 
 
 If we proceed to form the plane figure representing this 
 group, the sum of the internal angles of the quadrilateral 
 AiA^A^A^ is equal to four right angles, and the four circles 
 that form it therefore pass through a point. If this point be 
 taken at infinity, the four circles (and therefore all the circles of 
 the figure) become straight lines. The plane figure will now 
 take the form given in fig. 14, and the four generating opera- 
 tions are actual rotations through two right angles about lines 
 through Ai, A^, As and J.4, perpendicular to the plane of the 
 figure. 
 
294 
 
 GROUPS 
 
 [205 
 
 S2S1 
 
 SjSsSj 
 
 Fig. 14. 
 
 Every operation of the group is therefore, in this form of 
 representation, either a rotation through two right angles 
 about a corner of the figure or a translation ; and it will clearly 
 be the former or the latter according as it consists of an odd 
 or an even number of factors, when expressed in terms of the 
 generating operations. The operations which correspond to 
 translations form a sub-group; for if two operations each 
 consist of an even number of factors, so also does their 
 product. Moreover, this sub-group is self-conjugate, since the 
 number of factors in ^~^81 is even if the number in 8 is 
 even. This self-conjugate sub-group is generated by the two 
 operations 
 
 S1S2 and S2S3; 
 
 for SA = SA.8A, 
 
 SA = (Si82)~\ S3S2 = (SiSs)'^, 8sSi = (828^)'^ (SiS^y^ ; 
 and therefore every operation containing an even number of 
 factors can be represented in terms of 8182 and 8283. Lastly, 
 these two operations are permutable with each other ; for 
 
 8283 . 81S2 = 8283 . 8483 = 81S3 = 8182 . 8283 ; 
 
 and therefore every operation of the group is contained, once 
 and only once, in the form 
 
 S^'^{8,82y(8283)y, (a = 0, 1: /3, 7 = -<»,..„ 0, ..., 00). 
 
 The results thus arrived at may also be verified very simply 
 by purely kinematical considerations. If a group generated 
 
205] OF GENUS ONE 295 
 
 by Si, 82, Sg and Si is of finite order, there must, since it is of 
 genus 1, be either one or two additional relations between the 
 generating operations ; and any such relation is expressible by 
 equating the symbol of some operation of the general group to 
 unity. Such a relation is therefore either of the form 
 
 s,(SAf(s,s,y:=:i, 
 or (SAyiSAy^i. 
 
 The operation Si(SiS2y'{S2Ssy of the general group, con- 
 sisting of an odd number of factors, must be a rotation round 
 some corner of the figure, say a rotation round the corner A^ of 
 the white quadrilateral S; it is therefore identical with %~^Sr'2. 
 
 Now the relation S^'/S^S = 1, 
 gives Sr=l. 
 
 A relation of the first of the two forms is therefore incon- 
 sistent with the supposition that the group is actually generated 
 by Si, S2 and S3. It, in fact, reduces the generating operations 
 and the relations among them to 
 
 Si'^l, S,'=l, /Sf/ = 1, SAS, = 1, 
 
 which define a group of genus zero. 
 
 The only admissible relations for a group of genus 1 are 
 therefore those of the form 
 
 (SAy{SAy=i. 
 
 A single relation of this form reduces the operations of the 
 general group to those contained in 
 
 Si^(SAy(SA)y, 
 
 (a = 0, 1; /S = 0, 1, ...,6-1; 7=-x, ..., 0, ...,«); 
 and the group so defined is still of infinite order. 
 Finally, two independent relations 
 
 (S,s,f{SAy = i, 
 {SAf{SAY = i, 
 
 where p=l=J, 
 
 must necessarily lead to a group of finite order. If m is the 
 greatest common factor of b and b', so that 
 b = b^m, b' = bi'm, 
 
296 
 
 GROUPS 
 
 [205 
 
 where 61 and 6/ are relatively prime ; and if 
 
 61a; - 6/2/ = 1; 
 the two relations give 
 
 (SAy'^'-"'"^ = 1, 
 
 and (SAT = (SAY"-'''- 
 
 Every operation of the group is now contained, once and 
 only once, in the form 
 
 s,-{SAy{SA)\ 
 
 (a = 0, 1; /3 = 0, 1, ...,m-l; 7 = 0, 1, ..., cb^' -b,c' -1); 
 and the order of the group is 2 {be' ~ b'c). 
 
 206. Corresponding to the solution 
 
 n, = 3, mi = m2 = ms=S, 
 we have the general group generated by Si, 82, S3, where 
 S/=l, Si = l, -S/=l, 
 SiSA = 1. 
 The sum of the three angles of the triangle A^A^A, is two 
 
 Fig. 15. 
 
206] OF GENUS ONE 297 
 
 right angles, and therefore again the circles in the plane figure 
 (fig. 15) may be taken as straight lines. When the figure is 
 thus chosen, the generating operations are rotations through |7r 
 about the angles of an equilateral triangle ; and every operation 
 of the group is either a translation or a rotation. 
 
 The three operations 
 
 S1S2, S2S1S2, S2S1, 
 when transformed by S^, are interchanged among themselves. 
 When transformed by *Sfi, they become 
 
 8 'Si, Si^S^SiS^Si, Si' 82^81^, 
 and since {8182)^ = 1, 
 
 the two latter are 8^82' and 828182 respectively. 
 
 Hence the three operations generate a self-conjugate sub- 
 group ; and since 
 
 8182'- 828182 ■ 82^81 = 1, 
 
 this sub-group is generated by 81S2' and 828182- 
 
 These two operations are permutable ; for 
 
 8182'- 82S1O2 = O1O2 ^ 8281O2O1O2 . 02 ^ 828182 . 8182'. 
 
 Hence finally, every operation of the group is represented, 
 once and only once, by the form 
 
 Si-(SAy(S28i82)y, (a = 0, 1, 2; /3, 7= - 00 , ..., 0, ..., 00 ). 
 
 This result might also be arrived at by purely kinematical 
 considerations ; for an inspection of the figure shews that the 
 two simplest translations are 
 
 8182' and 828182, 
 and that every translation in the group can be obtained by the 
 combination and repetition of these. Every operation in which 
 the index a is not zero must be a rotation through |7r or ^tt 
 about one of the angles of the figure ; it is therefore neces- 
 sarily identical with an operation of the form 
 
 If now the group generated by *S^i, 82, 8^ is of finite order, 
 there must be either one or two additional relations between 
 them. A relation of the form 
 
 81- (8182')" {828182)" = !, 
 
298 GROUPS [206 
 
 where a is either 1 or 2, is equivalent to 
 
 2-^^,«S = 1, 
 so that Sr=\. 
 
 Such a relation would reduce the group to a cyclical group of 
 order 3. This is not admissible, if the group is actually to be 
 generated by two distinct operations 8-^ and 8^. 
 
 A relation (SA'f (8,8, 8,)' = 1, 
 
 gives, on transformation by S,"^, 
 
 (SA8,f(8,'8,y = l. 
 Now 8i'S, = {8,8^r (S.8A)-\ 
 
 so that (8 A')-' (8,8 Ay-' = 1. 
 
 If m is the greatest common factor of b and c, so that 
 b = b'm, c = c'm, 
 where b' and c are relatively prime ; and if 
 
 b'x — c'y = 1, 
 the two relations 
 
 (8A'y(8A8,y = i, 
 
 and (8A')-' (8,818,)"-"= I, 
 
 lead to (8.,8Ay^""-''''''+<'"> = 1, 
 
 and (8Ay" = (8,8 AT'^^'-"'' ^'"'"^ ; 
 
 and every operation of the group is contained, once and only 
 
 once, in the form 
 
 8,'^(8Ar(8,8A)', 
 where 
 
 a = 0, 1,2;^ = 0, l,...,m-l;'y=0,l, ...,m(6'^-6'c' + c'=)-l. 
 
 Thus the group is of finite order 3 (6^ — bc + c'). In this case 
 then, unlike the previous one, a single additional relation is 
 sufficient to ensure that the group is of finite order. Any 
 further relation, which is independent, must of necessity reduce 
 the group to a cyclical group of order 3 or to the identical 
 operation. 
 
 207. The two remaining solutions may now be treated in 
 less detail. The general group corresponding to the solution 
 
 n = 3, mi = 2, m, = 7113 = 4, 
 
207] 
 
 is given by 
 
 OF GENUS ONE 
 
 299 
 
 81828^ = 1, 
 and is represented graphically by fig. 16. 
 
 Pig. 16. 
 All the translations of the group can be generated from the 
 two operations 
 
 8182^, 8 2 81 82', 
 
 and every operation of the group is given, once and only once, 
 by the form 
 
 S2'^{8Ar{828,S2)y, (a = 0, 1,2,3; ;8, 7 = -oo , ...,0, ..., ^ ). 
 An additional relation of the form 
 
 82''{8,82'f(SAS2y^l, 
 where a is 1, 2 or 3, leads either to 
 
 S, = l, 82=1 or ^3 = 1, 
 and is therefore inconsistent with the supposition that the 
 group is generated by two distinct operations. 
 An additional relation 
 
 (8 A')" (828,82)" = 1 
 gives (8,82')-" (828,82)" =1: 
 
 and if b = him, c = Cim, 
 
 h,x + c-,y = 1, 
 
300 
 
 GROUPS 
 
 [207 
 
 where &i and Ci are relatively prime, these relations are 
 equivalent to 
 
 (8,8^)"" = (S,8A)"'^^'y-''^''K 
 
 Every operation of the group is then contained, once and 
 only once, in the form 
 
 8,'^(8^8,n8,8A)y, 
 
 (a = 0, 1, 2, 3; ;S = 0, 1, ..., m- 1; y = 0,l, ..^mibi' + c,')-!); 
 
 and the order of the group is 4 {¥ + c"), 
 
 208. Lastly, the general group corresponding to the 
 solution 
 
 m = 3, mi = 2, ma = 3, m^ = 6, 
 
 is given by 8,^=1, 8,'=1, 8,' = 1, 
 
 818283 = 1 ; 
 
 and it is represented graphically by fig. 17. 
 
 Fig. 17. 
 
209] OF GENUS ONE 301 
 
 Now it may again be verified, either from the generating 
 relations or from the figure, that the two operations 
 
 S.'SJ' and SA'Ss, 
 which are permutable with each other, generate all the opera- 
 tions which in the kinematical form of the group are translations ; 
 and that every operation of the group is represented, once and 
 only once, by the form 
 
 (a = 0,- 1, ..., 5; ,8, 7=-oo, ..., 0, ..., oo). 
 
 Also as before, any further relation, which does not reduce 
 the group to a cyclical group, is necessarily of the form 
 
 On transforming this relation by S^r^, we obtain 
 
 (SA'S,y(s,' 8,^)^ = 1. 
 
 Now S.J'S,' = (S,'S,T' (SsS,' S,) ; 
 
 so that (S^^S^^)-" (SASs)"*'' = 1- 
 
 If then b = bim, c = CiTti, 
 
 hx + c^y = 1, 
 where h^ and Ci are relatively prime, it follows that 
 
 and {S^^Siy = {S,8AT''^'y+'^ '"-*'(. 
 
 Every operation of the group is then contained, once and 
 only once, in the form 
 
 (a=0,l, ..., 5; ,5=0,1,. ..,m-l; 7 = 0,1, ...,m(6iH&iCi+Ci^)-l); 
 and the order of the group is 6 Qf + hc + &). 
 
 209. There are thus four distinct classes of groups' of 
 genus 1, which are defined in terms of their generating opera- 
 tions by the following sets of relations : — 
 
 I /SY = i, -5,^-1, s/ = i, {a,s,8,y = i, 
 
 (SAT (SA)' = 1, (SAY (SAf = 1, {ab' - a'b > 0); 
 N=2(ab'-a'b). 
 
 ' Dyck, "Ueber Aufstellung und Untersucliung von Gruppe und Irrationalitat 
 regularer Eiemann'scher Flaohen," Math. Ann. xvii, (1880), pp. 501 — 509. 
 
302 A GROUP [209 
 
 II. s,' = i, -sf/ = i, (SrS,y = i, 
 
 III. fifi^=l, /Sf/ = 1, (Si>SO^ = l, 
 
 {SA^Y{SA8;f> = \; 
 
 IV. ,Sf,'' = l, 8^ = 1, {8,8,y=l, 
 
 (S,^8^y(SA^8,)'' = l; 
 iV=6(a^ + a6 + 6^). 
 
 For special values of a and b, some of these groups may be 
 groups of genus zero; for instance, in Class I, if ab' — a'b is a 
 prime, the group is a dihedral group. It is left as an exercise 
 to the reader to determine all such exceptional cases. 
 
 Ex. Prove that the number of distinct types of group, of genus 
 two, is three ; viz. the groups defined by 
 
 (i) ^' = 1, B' = A% B-'AB = A-'; 
 
 (ii) ^^=1, B^=\, B-'AB = A'; 
 
 (iii) A^ = l, J5»=l, (^-8)3=1, {A'By=l. 
 
 210. As a final illustration of the present method of 
 graphical representation, we will consider the simple group of 
 order 168 (§ 146), given by 
 
 {(1236457), (234) (567), (2763) (45)}. 
 
 The operations of this group are of orders 7, 4, 3 and 2 ; and 
 it is easy to verify that three operations of orders 2, 3, and 7 
 can be chosen such that their product is identity. 
 
 In fact, if 
 
 >Sf2 = (16)(34), 6'3 = (253)(476), /S, = (1673524); 
 
 then 8,8,8, = 1. 
 
 Moreover, these three operations generate the group. The 
 connectivity of the corresponding surface, by the regular division 
 of which the group can be represented, is 2^ + 1 ; where 
 
 2^-2 = 168(3-2-i-J-|). 
 
210] 
 
 OF GENUS THREE 
 
 303 
 
 This gives P = 3 ; 
 
 it follows from Theorem II, § 197, that the genus of the 
 group is 3. 
 
 The figure for the general group, generated by S^, S^, and 
 
 St, where 
 
 S,'=l, 5f/=l, S,'=l, 
 
 S-iS^Sj = 1, 
 acquires as symmetrical a form as possible, by taking the 
 centre of the orthogonal circle for that angular point of the 
 triangle 1 at which the angle is ^tt. In fig. 18 a portion of 
 the general figure, which is contained between two radii of the 
 orthogonal circle inclined at an angle ^tt, is shewn. The 
 remainder may be filled in by inversions at the different 
 portions of the boundary. 
 
304 
 
 A GROUP 
 
 [210 
 
 The operations, which correspond to the white triangles of 
 the figure, are given by the following table : — 
 
 1 
 
 1 
 
 9 
 
 SA'S, 
 
 17 
 
 SA'S^Sr'S, 
 
 2 
 
 s. 
 
 10 
 
 OyOa^S'y^/S'a 
 
 18 
 
 S,%S,'S, 
 
 3 
 
 O7O2 
 
 11 
 
 07 02^7 02 
 
 19 
 
 SA% 
 
 4 
 
 S,^S, 
 
 12 
 
 O7 02*J7''02 
 
 20 
 
 SAS/S, 
 
 5 
 
 s/s. 
 
 13 
 
 S/SA'S, 
 
 21 
 
 SA'S, 
 
 6 
 
 87*^2 
 
 14 
 
 SA's, 
 
 22 
 
 Oy O2O7 O2 
 
 7 
 
 s/s. 
 
 15 
 
 0702*^7 '^a 
 
 23 
 
 S,%Sr% 
 
 8 
 
 Sr'S, 
 
 16 
 
 Sj 02*^7 ^2 
 
 24 
 
 8287^8287^82 
 
 The representation of the special group is derived from 
 this general figure by retaining only a set of 168 white (and 
 corresponding black) triangles, which are distinct when S^, S, 
 and *Si7 are replaced by the corresponding substitutions given on 
 p. 302. When each white triangle is thus marked with the 
 corresponding substitution, it is found that a complete set of 
 168 distinct white (and black) triangles is given by the portion 
 of the figure actually drawn and the six other distinct portions 
 obtained by rotating it round the centre of the orthogonal 
 circle through multiples of |7r. 
 
 To complete the graphical representation of the group of 
 order 168, it is necessary to determine the correspondence in 
 pairs of the sides of the boundary. This is facilitated by 
 
 noticing that the angular points ^1, A2, A^, of the 
 
 boundary must correspond in sets. Now the white triangle, 
 which has an angle at A-^ and lies inside the polygon, is 
 given by 
 
 This must be equivalent to a white triangle, which lies 
 outside the polygon, and has a side on the boundary and an 
 angle at one of the points A^, A^ 
 
 The triangle, which satisfies these conditions and has an 
 angle at -dara+i, is given by 
 
 while the triangle, which satisfies the conditions and has an 
 angle at ^sn+s. is given by 
 
210] OF GENUS THREE 305 
 
 When (16) (34) and (1673524) are written for S^ and S„ we 
 find that 
 
 The white triangle with an angle at A^ inside the polygon 
 is therefore equivalent to the white triangle with an angle at 
 An, which lies outside the polygon and has a side on the 
 boundary. It follows, from the continuity of the figure, that 
 the arcs A^A^ and A^Ai^ of the boundary correspond. Since 
 the operation 8j changes the figure and the boundary into 
 themselves, it follows that AgA^, AjsA^^', -^s-^e, -^i-^u', -^t^s, 
 A3A2', A^AiQ, A^A^; A11A12, .0.7^18; and ^i3.fl.i4, ^9^8 ; are pairs 
 of corresponding sides. Hence the above single condition is 
 sufficient to ensure that the general group shall reduce to the 
 special group of order 168. 
 
 By taking account of the relation 
 
 {S,S,f=l, or (S,s,y=i, 
 
 the form of the condition may be simplified. Thus it may be 
 written 
 
 S/ S2S/ S^ = O 2 07/3207 . 0'7^ySi2'S'7^'Si2*Si7° 
 
 or >S'2,Sf7^;Sf2 = 8/ 8,8/ S A' 8^8/. 
 
 Now 8,8, . 8,8, = 8,'8,8,<^8, . 8,8/8,8/ 
 
 = 8/8,8/8,8/. 
 
 Hence 8/8,8/8,8, = 8,8/8,8/8,, 
 or 8,*8,8/ . 8,8/8, = 8,8/8,8/8,, 
 
 or 8/8,8/8,8/^8,8/8,, 
 
 or finally (8/8,)*=!. 
 
 The simple group of order 168 is therefore defined abstractly 
 by the relations' 
 
 8/ = l, S/=l, (8,8,y=l, {8/8,y = l. 
 
 Ex. Shew that the symmetric group of degree five is a group 
 of genus four ; and that it is completely defined by the relations 
 
 1 This agrees with the result as stated by Dyck, " Gruppentheoretische 
 Studien," Math. Ann. Vol. xx, (1882), p. 41. 
 
 B. 
 
 20 
 
306 cayley's [211 
 
 211. The regular division of a continuous surface into 
 2N black and white polygons is only one of many methods that 
 may be conceived for representing a group graphically. 
 
 We shall now describe shortly another such mode of 
 representation, due to Cayley^ who has called it the method 
 of colour-groups. As given by Cayley, this method is entirely 
 independent of the one we have been hitherto dealing with; 
 but there is an intimate relation between them, and the new 
 method can be most readily presented to the reader by deriving 
 it from the old one. 
 
 Let l,S„S„ ,;Sf^_, 
 
 be the operations of a group G of order JSf. We may take the 
 N— 1 operations other than identity as a set of generating 
 operations. Their continued product 
 
 'S'l'Sfg ^N-l 
 
 is some definite operation of the group. If it is the identical 
 operation, the only modification in the figure, which represents 
 the group by the regular division of a continuous surface, will 
 be that the iVth corner of the polygon has an angle of two right- 
 angles. 
 
 With this set of generating operations, the representation 
 of the group is given by a regular division of a continuous 
 surface into M white and N black polygons A^A^ ^j^r, the 
 
 angle at Ar being — , m^ being the order of Sr. Suppose 
 
 now that in each white polygon we mark a definite point. 
 From the marked point in the polygon 2, draw a line to 
 the marked point in the polygon derived from it by a positive 
 rotation round its angle A^. Call this line an ^^-line, and 
 denote the direction in which it is drawn by means of an 
 arrow. Carry out this construction for each polygon 2, and 
 for each of its angles except A^f. We thus form a figure 
 which, disregarding the original surface, consists of N points 
 connected by JH (iV — 1) directed lines, two distinct lines 
 joining each pair of points. Now if the line drawn from a 
 
 > American Journal of Mathematics, Vol. i, (1878), pp. 174—176, Vol. xi, 
 (1889), pp. 139 — 157 ; Proceedings of the London Mathematical Society, Vol. ix, 
 (1878), pp. 126—133. 
 
212] COLOUR GROUPS 307 
 
 to 6, where a and 6 are two of the points, is an >Sf-line, then 
 the line drawn from & to a is from the construction an (S^'-line. 
 We may then at once modify our diagram, in the direction 
 of simplification, by dropping out one of the two lines 
 between a and h, say the S~'-Iine, on the understanding that 
 the remaining line, with the arrow-head reversed, will give the 
 line omitted. If /S is an operation of order 2, B and S~^ are 
 identical, and two arrow-heads may be drawn on such a line in 
 opposite directions. The modified figure will now consist of 
 N points connected by \N{N — 1) lines. From the construction 
 it follows at once that, for every value of ?■, a single (S^-line ends 
 at each point of the figure and a single /S^-line begins at each 
 point of the figure ; these two lines being identical when the 
 order of /S^ is 2. 
 
 We may pass from one point of the figure to another along 
 the lines in various ways ; but any path between two points of 
 the figure will be specified completely by such directions as: 
 follow first an jSf^-line, then an )Sj-line, then an /S(~^-line, and 
 so on. Such a set of directions is said to define a route. 
 It is an immediate consequence of the construction that, if 
 starting from some one particular point a given route leads back 
 to the starting point, then it will lead back to the starting point 
 from whatever point we begin. In fact, a route will be specified 
 symbolically by a symbol 
 
 ^u Sf^SsSr, 
 
 and if ;Sf„ St-^S,SrX = t, 
 
 then ,S„ Sr'SsSr = l, 
 
 and therefore >Sf„ Sr'SsSrl,' = %', 
 
 whatever operation 2' may be. 
 
 212. If the diagram of N points connected by ^N(N— 1) 
 directed lines is to appeal readily to the eye, some method must 
 be adopted of easily distinguishing an /S^-line from an /Sj-line. 
 To effect this purpose, Cayley suggested that all the /S^-lines 
 should be of one colour, all the ;Sg-lines of another, and so on. 
 Suppose now that, independently of any previous consideration, 
 we have a diagram of iV points connected by ^N(N — l) coloured 
 directed lines satisfying the following conditions : — 
 
 20—2 
 
308 cayley's [212 
 
 (i) all the lines of any one colour have either (a) a single 
 arrow-head denoting their directions : or (b) two opposed arrow- 
 heads, in which case each may be regarded as equivalent to two 
 coincident lines in opposite directions ; 
 
 (ii) there is a single line of any given colour leading to every 
 point in the diagram, and a single line of the colour leading from 
 every point : if the colour is one with double arrow-heads, the 
 two lines are a pair of coincident lines ; 
 
 (iii) every route which, starting from some one given point 
 in the diagram, is closed, i.e. leads back again to the given point, 
 is closed whatever the starting point. 
 
 Then, under these conditions, the diagram represents in 
 graphical form a definite group of order N. 
 
 It is to be noticed that the first two conditions are necessary 
 in order that the phrase " a route " used in the third shall have 
 a definite meaning. Suppose that R and R' are two routes 
 leading from a to b. Then RR'-^ is a closed route and will 
 lead back to the initial point whatever it may be. Hence if R 
 leads from c to d, so also must R' ; and therefore R and R' are 
 equivalent routes in the sense that from any given starting 
 point they lead to the same final point. There are then, with 
 identity which displaces no point, just N non-equivalent routes 
 on the diagram, and the product of any two of these is a definite 
 route of the set. The N routes may be regarded as operations 
 performed on the iV points ; on account of the last property 
 which has been pointed out, they form a group. Moreover, the 
 diagram gives in explicit form the complete multiplication table 
 of the group, for a mere inspection will immediately determine 
 the one-line route which is equivalent to any given route ; i.e. 
 the operation of the group which is the same as the product of 
 any set of operations in any given order. 
 
 From a slightly different point of view, every route will give 
 a permutation of the iV points, regarded as a set of symbols, 
 among themselves; no symbol remaining unchanged unless 
 they all do. To the set of N independent routes, there will 
 correspond a set of iV" substitutions performed on N symbols ; 
 and we can therefore immediately from the diagram represent 
 the group as a transitive substitution group of degree N. 
 
214] COLOUR GROUPS 309 
 
 213. It cannot be denied that, even for groups of small 
 order, the diagram we have been describing would not be easily 
 grasped by the eye. It may however still be considerably 
 simplified since, so far as a graphical definition of the group 
 is concerned, a large number of the lines are always redundant. 
 
 If in the diagram consisting of iV points and ^N{N— 1) 
 coloured lines, which satisfies the conditions of § 212, all the 
 lines of one or more colours are omitted, two cases may occur. 
 We may still have a figure in which it is possible to pass 
 along lines from one point to any other; or the points may 
 break up into sets such that those of any one set are con- 
 nected by lines, while there are no lines which enable us 
 to pass from one set to another. 
 
 Suppose, to begin with, that the first is the case. There will 
 then, as before, be iV" non-equivalent routes in the figure, which 
 form a group when they are regarded as operations ; it is 
 obviously the same group as is given by the general figure. 
 The sole difference is that there will not now be a one-line 
 route leading from every point to every other point, and there- 
 fore the diagram will no longer give directly the result of the 
 product of any number of operations of the group. 
 
 If on the other hand the points break up into sets, the new 
 diagram will no longer represent the same group as the original 
 diagram. Some of the routes of the original diagram will 
 not be possible on the new one, but every route on the new 
 one will be a route on the original diagram. Hence the new 
 diagram will give a sub-group ; and since it is still the case that 
 no route, except identity, can leave any point unchanged, the 
 number of points in each of the sets must be the same. The 
 reader may verify that the. sub-group thus obtained will be self- 
 conjugate, only if the omitted colours interchange these sets 
 bodily among themselves. 
 
 214. The simplest diagram that will represent the group 
 will be that which contains the smallest number of colours and 
 at the same time connects all the points. To each colour 
 corresponds a definite operation of the group (and its inverse). 
 Hence the smallest number of colours is the smallest number 
 of operations that will generate the group. It may be noticed 
 
310 cayley's colour groups [214 
 
 that this simplified diagram can be actually constructed from the 
 previously obtained representation of the group by the regular 
 division of a surface, the process being exactly the same as 
 that by which the general diagram was obtained. For if 
 
 are a set of independent generating operations, and if 
 
 we may represent the group by the regular division of a surface 
 into 2iV" black and white (n + l)-sided polygons. When we 
 
 draw on this surface the 8i-, 82-, , *S„-lines, the N points 
 
 will be connected by lines in a single set, since from 
 
 every operation of the group can be constructed ; and the set 
 of points and directed coloured lines so obtained is clearly the 
 diagram required. 
 
 As an illustration of this form of graphical representation, 
 we may consider the octohedral group (§ 201), defined by 
 
 Si 82 8s = 1. 
 On the diagram already given (p. 290), we may at once draw 
 81-, 83- and iSs-lines. These in the present figure are coloured 
 respectively red, yellow, and green. On the lines which corre- 
 spond to operations of order two (in this case the red Hnes), 
 the double arrow-heads may be dispensed with. By omitting 
 successively the red, the yellow, and the green lines we form 
 from this the three simplest colour diagrams which will re- 
 present the groups 
 
 1 For further illustrations, the reader may refer to Young, Amer. Journal, 
 Vol. XV, (1893), pp. 164—167; Maschke, Avier. Journal, Vol. xvui, (1896), 
 pp. 156—188. 
 
To face page 310. 
 
CHAPTER XIV. 
 
 ON THE LINEAR GROUP \ 
 
 215. We shall now, in illustration of the general principles 
 that have been developed in the preceding chapters, proceed to 
 discuss and give an analysis of certain special groups. The first 
 that we choose for this purpose is the group of isomorphisms 
 
 of an Abelian group of order p™ and type (1, 1, to n units). 
 
 This group has been defined and its order determined in 
 §§ 171, 172. It is there shewn that the group is simply 
 isomorphic with the homogeneous linear group defined by all 
 sets of congruences 
 
 2/2 ^= ^21 ^1 ~r ^22 ^2 "T~ ~r ^Sft '^n ' / J \ 
 
 IJn = ^ni^i "T C(ji2^2 T "r Cinn^U} 
 
 ' The homogeneous linear group and its sub-groups forms the subject of the 
 greater part of Jordan's Traite des Substitutions. The investigation of its 
 composition-series, given in the text, is due to Jordan. 
 
 The complete analysis of the fractional linear group, defined by 
 
 ax + 8 , , ^ 
 
 where ad-^ysl, 
 
 is due originally to Gierster, " Die Untergruppeu der Galois'schen Gruppe der 
 
 Modulargleichungen fur den Fall eines primzahligea Transformationsgrades," 
 
 Math. Ann. Vol. xviii, (1881), pp. 319—365. With a few unimportant 
 
 modifications, the investigation in the text follows the lines of Gierster'a 
 
 memoir. 
 
 A similar analysis of the simple groups of order 2''(2^"-l), which can be 
 expressed as triply transitive groups of degree 2"-|-l, has been given by the 
 author, "On a class of groups defined by congruences," Proc. L. M. S. Vol. xxv, 
 (1894), pp. 132—136. 
 
312 THE HOMOGENEOUS [215 
 
 whose determinants are not congruent to zero; and that its 
 order is 
 
 N = {f - V) {p'" - p) (p» -^»-'). 
 
 The operation given by the above set of congruences ■will be 
 denoted in future by the symbol 
 
 amOOi-\- +«n»«)i). 
 
 216. It may be easily verified by direct calculation that, if 
 D and D' are the determinants of two operations 8 and S' of 
 the group, then DD' is the determinant of the operation 8S', 
 all the numbers involved being reduced, mod. p. Hence it 
 immediately follows that those operations of the group, whose 
 determinant is unity, form a sub-group. If this sub-group is 
 denoted by V, the group itself being G, then F is a self- 
 conjugate sub-group of G. For if 2 is any operation of V and 
 S any operation of G whose determinant is D, the determinant 
 of S~^XS is D~^D or unity ; and therefore S~^'^S belongs to F. 
 Suppose now that S is an operation^ of G whose determinant is 
 z, a primitive root of the congruence 
 
 ^3)-i = 1 (mod. p). 
 
 Then the determinant of every operation of the set 
 
 ST 
 
 is z'' ; and therefore, if r and s are not congruent (mod. p), the 
 
 two sets 
 
 ST and ST 
 
 can have no operation in common. Moreover, if S' is any 
 operation of G whose determinant is zf^, then S-^S' belongs to 
 F, and therefore S' belongs to the set ST. Hence finally, the 
 
 sets 
 
 r,sr,8T, ,Sp-T 
 
 are all distinct, and they include every operation of (? ; so that 
 
 G = {S,T}. 
 
 C 
 The factor-group ^ is therefore cyclical and of order p — l. 
 
 ' The operation [zx^ , Xj , . . . , .t„) has z for its determinant. 
 
217] LINEAR GROUP 313 
 
 217. It may be very readily verified that the operations of 
 the cyclical sub-group generated by 
 
 are self-conjugate operations of 0. To prove that these are 
 the only self-conjugate operations of G, we will deal with the 
 case w = 3: it will be seen that the method is perfectly 
 general. Suppose then that 
 
 T = {axi + ^Xi + <yx3, a'xi + lS'x^ + y'x,, a"Xi + ^"Xi + '^"x^ 
 
 is a self-conjugate operation of 0, while 
 
 S = (axi-\-hx^-\-cxs, a'x-^ + h'x^ + c'x^, a"xi + h"x2 + c"x^ 
 
 is any operation. The relation 
 
 8T=TS 
 
 involves the nine simultaneous congruences' 
 
 aa. -f- ha' + ca!' s ota -|- ^a' -I- 7a", 
 
 ajS + 6/3' + c0" = ab+ ^V + yb", 
 
 ay + by + cy" = ac -f- ^c + yc", 
 
 etc., etc. ; 
 
 and these must be satisfied for all possible values of the 
 coefiicients of S. Now 
 
 6 = c = a' = 
 
 is a possible relation between the coefficients of 8, whether 
 regarded as an operation of ff or V; and therefore 
 
 7 = 0. 
 
 In the same way, it may be shewn that 
 
 /3 = a' = y' = a" = fi"=0, 
 and that a = /3' s 7" ; 
 
 so that r is a power of the operation 
 
 {zXi, ZX2, ZX3). 
 
 The only self-conjugate operations of G are therefore the powers 
 of A, where A denotes 
 
 \ZXi, ZX^, , ZXji)', 
 
 and the only self-conjugate operations of T are those operations 
 
 ' These and all succeeding congruences are to be taken mod. p, unless the 
 contrary is stated. 
 
314 THE HOMOGENEOUS [217 
 
 of this cyclical sub-group which are contained ia F. Now the 
 order of ^ is p — 1 and its determinant is z": Hence the self- 
 conjugate operations of V form a cyclical sub-group D of order 
 
 d, where d is the greatest common factor of ^ — 1 and n ; and 
 
 p-i 
 
 this sub-group is generated by ^ "^ . 
 
 218. To determine completely the composition-series of G, 
 it is necessary to find whether T has a self- conjugate sub- 
 group greater than and containing D. A simple calculation 
 will shew that, from 
 
 and its conjugate operations, all the operations of T may be 
 generated ; and hence no self- conjugate sub-group of F which 
 is different from F itself can contain an operation of this form. 
 If then it is shewn that any self-conjugate sub-group of F, 
 distinct from D, necessarily contains operations of this form, it 
 follows that D is a maximum self-conjugate sub-group of F. 
 
 We shall first deal with the case n = 2. 
 
 If p = 2, the orders of G, F and D are 6, 6 and 1. In this 
 case, F is simply isomorphic with the symmetric group of three 
 symbols, which has a self-conjugate sub-group of order 3. The 
 successive factor-groups of the composition-series of are 
 therefore cyclical groups of orders 2 and 3. 
 
 li p = S, the orders of G, F and B are 48, 24 and 2. The 
 p 
 factor-group j^ has 12 for its order, and cannot therefore be a 
 
 simple group. The reader will have no difficulty in verifying 
 that, in this case, the successive factor-groups of G have orders 
 2, 3, 2, 2 and 2. We may therefore, in dealing with the case 
 71 = 2, assume that p is not less than 5. 
 
 Let us suppose now that F has a self-conjugate sub-group 
 / that contains D ; and let S or 
 
 (axi + hx^, a'x-y -\- b'x^) 
 be one of its operations, not contained in D. 
 
 If 6 is different from zero, F contains S, where 1, denotes 
 
 / , , l-haV \ 
 
 I 'MXi + Mx^, -, — «i — aax^) , 
 
219] LINEAR GROUP 315 
 
 and therefore / contains 1,~'^8'2,S, which is 
 
 ,-.^ (1 + a') {V + aoe) 
 ha? 
 
 -a 'oc^,- -i^„ ah - c^x. 
 
 If h is zero, h' is congruent with ar^ ; therefore, in any case, 
 / contains an operation B' of the form 
 
 (cx-i, dxi + c~^x^. 
 
 Again, T contains the operation T, where T denotes 
 
 \Xi, Xi + X^)', 
 
 and I therefore contains S'T~^S'~^T, which is 
 («i, {l-c')xi + x^). 
 Hence unless 1 — c" s 0, / must coincide with T. Now, 
 when p>5, c can always be chosen so that this congruence 
 is not satisfied. If p=b, the square of the above operation 
 X'^SXS, when unity is written for a, is 
 
 {xi,-^(b' +a)xi + icA; 
 
 unless b' + a=0, this again requires that / coincides with T. 
 If finally, the condition b' + a = is satisfied in S, it is not 
 satisfied in ST~^S~^T, another operation belonging to /; and 
 therefore again, in this case, I coincides with F. 
 
 r 
 
 Hence finally, if w = 2, the factor-group -^ is simple, except 
 when ^ is 2 or 3. 
 
 219. When n is greater than 2, it will be found that it is 
 sufficient to deal in detail with the case n=S, as the method 
 will apply equally well for any greater value of n. Suppose 
 here again that F has a self-conjugate sub-group I which 
 contains D; and let 8, denoting 
 
 (axi + bx2 + cx„ a'xi + b'x^ -I- c'x^, a"xi + Vx^ -\- c"x^, 
 
 be one of the operations of I which is not contained in D. 
 8 cannot be permutable with all operations of the form 
 {x^, x^, Xn + x^, as it would then be permutable with every 
 operation of F. We may therefore suppose without loss of 
 generality that 8 and T are not permutable, T denoting 
 {xi, X2, X3 + X1). Then T'^8TS~^ is an operation, distinct from 
 
316 THE HOMOGENEOUS [219 
 
 identity, belonging to /. Now a simple calculation shews that 
 this operation, say U, is of the form 
 
 (a;i — cX, X, — c'X, Axi + Bx^ + Gx^, 
 
 where X is the symbol with which S"^ replaces x^. 
 
 If c and c' are both different from zero, F will contain an 
 operation V of the form 
 
 (■ 
 
 Xi / *2> *2 ) "'a ) ) 
 
 and / contains F^' UV, which is of the form 
 
 {x^, ax^ + /8«2 + 7^3, «'«! + /3'a!2 + '■^'x^. 
 
 Moreover, if either c or c' is zero, the operation U itself 
 leaves one symbol unaltered. Hence / always contains opera- 
 tions by which one symbol is unaltered. 
 
 This process may now be repeated to shew that I necessarily 
 contains operations of the form 
 
 («!, x^, a."xi + ^"Xi + 7"«3) ; 
 
 and, since the determinant of the operation is unity, 7" is 
 necessarily congruent to unity. But it has been seen that 
 the group F is generated from the last operation and the 
 operations conjugate to it. Hence finally, if n is greater than 
 
 F 
 
 2, the factor-group -^ is simple for all values of p. 
 
 220. The composition-series of G is now, except as regards 
 the constitution of the simple group j^ , perfectly definite. It 
 
 has, in fact, been seen that p and D are cyclical groups of orders 
 
 p-1 and d; and therefore if a, /8, 7, are distinct primes 
 
 whose product is p-1, and if a', y8', 7', are distinct primes 
 
 whose product is d ; the successive factor-groups of G are first, 
 
 a series of simple groups of prime orders a, /3, 7, : then 
 
 N 
 a simple group of composite order — _ , : and lastly, a series 
 
 of simple groups of prime orders a', /3', 7', 
 
 The sequence in which the set of simple groups of orders 
 
221] LINEAR GROUP 317 
 
 a, /3, 7, are taken in the composition-series may be clearly 
 
 any whatever, and the same is true of the set of factor-groups 
 
 of orders a', /8', 7', ; but it is to be noticed that, when d is 
 
 not equal to p — 1, the composition-series is capable of further 
 
 modifications. In this case, {A, T] is a self-conjugate sub-group 
 
 N 
 of of order -r, which has a maximum self-conjugate sub-group 
 
 [A] of order p—1. The successive composition-factors of G 
 may therefore be taken in the sequence 
 
 and their arrangement may be yet further changed by con- 
 sidering the self-conjugate sub-group [A'^, F}, where m is a 
 
 factor oi p — 1 less than ^^—j — . 
 
 221. For every value of p^, except 2" and 3^, it thus 
 appears that the linear group may be regarded as defining a 
 simple group of composite order. We shall now proceed 
 to a discussion of the constitution of the simple groups thus 
 defined when n = 2, p being greater than 3 *- In this case, the 
 group F is defined by the congruences 
 
 2/1 = our, -I- /3a;2, 
 2/2 = 7^1 + S«2, (mod. p) ; 
 aB-l3y = 1, 
 
 and since ^ — 1 is divisible by 2 when p is an odd prime, d is 
 equal to 2. Hence the self-conjugate operations of F are 
 
 («!, ajj) and (— «i, — a;^). 
 
 The order of F is p (p" — 1), and therefore the order of the 
 simple group, H, which it defines is ^^ ip^— !)• Suppose now, 
 if possible, that F contains a sub-group g simply isomorphic 
 with H. If 8 is any operation of F, not contained in g, the 
 whole of the operations of F are contained in the two sets 
 
 9,Sg. 
 
 * For the case m = 3 the reader may consult a paper by the author "On a 
 class of groups defined by congruences," Proc. L. M. S. Vol. xxvi, (1895), pp. 58 — 
 106. 
 
318 THE FRACTIONAL [221 
 
 Now (a-'a, —Xi), whose square (— «i, —x^) is a self-conjugate 
 operation, cannot be contained in the simple group g. Hence 
 both (w^, — Xi) and (— x^, — x^) are contained in Sg, an obvious 
 contradiction. Therefore F contaius no sub-group simply 
 isomorphic with H. 
 
 For a discussion of the properties of H, some concrete 
 representation of the group itself is necessary; this may be 
 obtained in the following way. Instead of the pair of homo- 
 geneous congruences that define each operation of F, let us, 
 as in § 113, consider the single non-homogeneous congruence 
 
 ax + . , , 
 
 where aS — ^y = 1. 
 
 Corresponding to every operation 
 
 (axi + ^x^, 7*1 -)- Saja) 
 of F, there will be a single operation of this new set ; namely 
 that in which a, ^, 7, 8 have respectively the same values. But 
 since the operations 
 
 ax + /3 , — ax — B 
 
 y = p and y= . 
 
 ^ ryx+B ^ —jx- 8 
 
 are identical, two operations 
 
 (axi + ^x^, 7^1 + SiPj) and (— a«i — ^x^, — yx^ — Sx^) 
 
 of F will correspond to each operation 
 
 _ cuc + ^ 
 
 y ~ ryx+B 
 
 of the new set ; the two self-conjugate operations 
 
 («!, X2) and (—«!,- x^), 
 
 in particular, corresponding to the identical operation of the 
 new set. Moreover, direct calculation immediately verifies 
 that, to the product of any two operations of F, corresponds the 
 product of the two corresponding operations of the new set. 
 Hence the new set of operations forms a group of order ^p (p' — 1), 
 with which F is multiply isomorphic ; the group of order 2 
 formed by the self-conjugate operations of F corresponding to 
 the identical operation of the new group. 
 
222] LINEAR GROUP 319 
 
 The simple group H, of order \p {p^ — 1). which we propose 
 to discuss, can therefore be represented by the set of operations 
 
 where ah — ^iy= \, 
 
 a, /3, 7, S being integers reduced to modulus p. 
 
 222. Since the order of H is divisible by p and not by p^, 
 the group must contain a single conjugate set of sub-groups of 
 order p. Now the operation 
 
 y = x+l, 
 
 or {x + 1) as we will write it in future, is clearly an operation 
 of order p : for its ?zth power is {x + n), and p is the smallest 
 value of n for which this is the identical operation. If 
 
 {x + 1) and ( ^J are represented by P and S, then 
 
 V-7^«-l-l+a7/ 
 
 This is identical witLP, only if 
 
 7 = 0, a^ = 1 ; 
 
 and therefore P is permutable with no operations except its 
 own powers. On the other hand, if 
 
 7 = 0, 
 
 then S-'P8 = P'--'; 
 
 and therefore every operation, for which 7 = 0, transforms the 
 sub-group [P] into itself. These operations therefore form a 
 sub-group : a result that may also be easily verified directly. 
 The order of this sub-group is the number of distinct operations 
 
 * for which ah = 1. The ratio ^ must be a quadratic 
 
 residue, while /3 may have any value whatever. Hence the 
 order of the sub-group i&\{p — l)p; and H therefore contains 
 p + 1 sub-groups of order p. Since jff is a simple . group, it 
 follows (I 125) that it can be represented as a transitive 
 substitution group of degree _p + l, 
 
320 THE FRACTIONAL [222 
 
 This representation of the group can be directly derived, as 
 in § 113, from the congruences already used to define it. Thus 
 if, in 
 
 _ aa; + jS 
 
 we write for x successively 0, 1, 2, , p — \, oo, the p + \ 
 
 values obtained for y, when reduced mod. p, will be the same 
 p-\-\ symbols in some other sequence. For if 
 
 aa^i + /3 _ 0^2 + /8 
 
 7a?i + h~ 'yXi + S' 
 
 then (aS — /37) (^i - x^) = 0, 
 
 and therefore x^sXi. 
 
 Each operation of H gives therefore a distinct substitution 
 
 performed on the symbols 0, 1, ,p-l,txi; and the complete 
 
 set of substitutions thus obtained gives the representation of H 
 as a transitive substitution group of degree ^ + 1. Since H 
 contains operations of order p, this substitution group must be 
 doubly transitive. That this is the case may also be shewn 
 directly. Thus 
 
 y — a X — a' 
 
 — T = '^ jT' 
 
 y — x — o 
 
 is an operation changing a' into a and b' into b. This operation 
 may be written 
 
 _ k (bm —a)x + k (ab' — ma'b) 
 y " 'k{m-\)x-\-k{b'-ma'Y ' 
 and its determinant is 
 
 A;"™ (6 -a) (6' -a'). 
 If now (6 — a) (b' — a') is a quadratic residue (or non-residue) 
 mod. p, m may be any quadratic residue (or non-residue) ; and k 
 can always be chosen so that the determinant is unity. There 
 are therefore \{p — 1) substitutions in the group, changing any 
 two symbols a , b' into any other two given symbols a, b. 
 
 ^j keeps X unchanged in the 
 
 substitution group, x must satisfy the congruence 
 
 _ ax + ^ 
 ''-yx + B' 
 
223] LINEAR GROUP 321 
 
 that is •yo(? + {h — a)x — ^ = 0. 
 
 Such a congruence cannot have more than two roots ; and 
 therefore every substitution displaces all, all but one, or all but 
 two, of the p+1 symbols. 
 
 223. The substitutions, which keep either one or two 
 symbols fixed, must therefore be regular in the remaining p or 
 p - 1 symbols. Hence the order of every substitution which 
 keeps just one symbol fixed must be p ; and the order of every 
 substitution that keeps two symbols fixed must be equal to or 
 be a factor oi p — \. Now it was seen in the last paragraph 
 that the order of the sub-group that keeps two symbols fixed is 
 \{p — 1). Moreover, if 5: is a primitive root mod. p, the 
 sub-group that keeps a and h fixed contains the operation 
 
 y — h x — h' 
 
 and the order of this operation is ^ (p — 1). Hence, the 
 sub-group that keeps any two symbols fixed is a cyclical group 
 of order ^ (^ — 1) ; and every operation that keeps two symbols 
 fixed is some power of an operation of order ^ (^ — 1). Since the 
 group is a doubly transitive group of degree ^ -|- 1, there must 
 be 4 (jo -F 1) p sub-groups which keep two symbols fixed ; and 
 these must form a conjugate set. Each is therefore self- 
 conjugate in a sub-group of order p—1. To determine the 
 type of this sub-group, we may consider the sub-group keeping 
 
 f zee \ 
 and X fixed : this is generated by Q, where Q denotes ( ^-J • 
 
 If {^ — ^j is represented by B, then 
 
 / (^ag-^'/37)a;-(5-0«/3 
 
 which can be a power of Q only if 
 
 a^ = 0, 78 = 0. 
 
 Hence either yS = 7 = 0, 
 
 in which case S is a power of Q : or 
 
 asSsO, 7 = -/8-^ 
 
 21 
 
322 THE FRACTIONAL [223 
 
 In the latter case, we have 
 which is an operation of order 2 ; and th6n 
 
 The group of order ^ — 1, which contains self-conjugately a 
 cyclical sub-group of order ^(p — 1) that keeps two symbols fixed, 
 is therefore a group of dihedral (§ 202) type. Moreover, if t is 
 any factor oi p—1, this investigation shews that {S, Q] is the 
 greatest sub-group that contains {Q*} self-conjugately. 
 
 224. A substitution that changes all the symbols must 
 either be regular in the p+1 symbols, or must be such that 
 one of its powers keeps two symbols fixed. The latter case 
 however cannot occur ; for we have just seen that, if Q is an 
 operation, of order ^{p — 1), which keeps two symbols fixed, 
 the only operations permutable with Q" are the powers of Q. 
 Hence the substitutions that change all the symbols must 
 be regular in the ^ -I- 1 symbols, and their orders must be 
 equal to or be factors ot p + 1. 
 
 Suppose now that i is a primitive root of the congruence 
 
 iP»-i -1 = (mod. p), 
 
 so that i and i^ are the roots of a quadratic congruence with 
 real coefficients ; and consider the operation K, denoting 
 
 where h is not a multiple of ^ + 1. On solving with respect to 
 y, K is expressed in the form 
 
 .{|+i)(P-i)_.-(^i)(i>-i) 
 
 
 y= 
 
 i(.-i)_i-(i>-i) _*(..,. ,-(|-0'*-i)_v-(l-O(^-i>' 
 
 -I- 
 
 .f(P-i)_.-f(^-i) J'^-^)_-i(^-« 
 
225] LINEAR GROUP 323 
 
 an operation of determinant unity. It will be found, on writing 
 iP for i in the coefficients of this operation, that they remain 
 unaltered; therefore, since they are symmetric functions of 
 i and i^, they must be real numbers. The operation therefore 
 belongs to H. The nth. power of this operation is given by 
 
 y — i^P X — i^P ' 
 
 and therefore, since the first power of i which is congruent to 
 unity, mod. p, is the {p^ — l)th, the order of the operation is 
 ^(jo + 1). If we write kp for k in the operation K, the new 
 operation is K~^ ; but if k is replaced by any other number k', 
 which is not a multiple of p + 1, the new operation K', given by 
 
 V — i''' .„,„ ,, « — i*" 
 
 y _ ^k'p g, _ ^k'p ' 
 
 generates a new sub-group of order ^(^ + 1), which has no 
 operation except identity in common with [K]. Now there are 
 p' — p numbers less than p^—1 which are not multiples of 
 y + 1 ; therefore H contains ^ (p'^ — p) cyclical sub-groups of 
 order ^(p + 1), no two of which have a common operation 
 except identity. The corresponding substitutions displace all 
 the symbols. 
 
 225. A simple enumeration shews that the operations of 
 the cyclical sub-groups of orders ^(p—1), p and ^(p + l), 
 exhaust all the operations of the group. Thus there are, 
 omitting identity from each sub-group: 
 
 (i) ^p {p + 1) sub-groups of order ^(p — 1), containing 
 ip{p^ — I) — ^p^— Ip distinct operations; 
 
 (ii) ^p{p— I) sub-groups of order i (jp + 1), containing 
 Ip (p^ — 1) — ^p^ + ^p distinct operations ; 
 
 (iii) p + 1 sub-groups of order p, containing 
 p^ — I distinct operations ; 
 
 and the sum of these numbers, with 1 for the identical opera- 
 tion, gives ^p {p^ — 1), which is the order of the group. 
 
 Every operation that displaces all the symbols is therefore 
 the power of an operation of order \{p+ 1). 
 
 21—2 
 
324 THE FRACTIONAL [226 
 
 226. We shall now further shew that the ^p (p — 1) sub- 
 groups of order | (p-l- 1) form a single conjugate set, and that 
 each is contained self-conjugately in a dihedral group of order 
 ^ + 1. Let S be any operation of H, which is permutable with 
 {K} and replaces i* by some other symbol j. Then 8~^K8 is 
 an operation which leaves j unaltered ; it may therefore be ex- 
 pressed in the form 
 
 y — j 00 — j 
 ■^ — ^ = m 4 ■ 
 
 y-j «-; 
 
 This can belong to the sub-group generated by K, only if j 
 and j' are the same pair as i* and i'^. Hence j must be either 
 i* or i^ ; and similarly, if S replaces i^P by /, the latter must be 
 either i'^ or i*. Hence either S_ must keep both the symbols 
 i* and i'^ unchanged or it must interchange them ; and con- 
 versely, every operation which either keeps both the symbols 
 unchanged or interchanges them, must transform {K} into 
 itself If S keeps both of them unchanged, it is a power of 
 K, If 8 interchanges them, it is of the form 
 
 V — i* OS — i^ 
 
 y — v'P x — i" 
 
 and a simple calculation shews that 
 
 8-'K8 = K-\ 
 
 If we take m = \, 8 becomes 
 
 an operation belonging to H. Hence the cyclical sub-group [K] 
 is contained self-conjugately in the sub-group of (^S, K] which is 
 of dihedral type. If there were any other operation 8', not con- 
 tained in {8, K], which transformed K into its inverse, then 
 88' would be an operation permutable with K and not con- 
 tained in [K]. It has just been seen that no such operation 
 exists. Hence {8, K], of order p -l- 1, is the greatest sub-group 
 that contains [K] self-conjugately; and [K] must be one of 
 hP (P ~ 1) conjugate sub-groups. 
 
 227. The distribution of the operations of H in conjugate 
 sets is now known. A sub-group of order p is contained self- 
 conjugately in a group of order ip(p — 1), while an operation 
 
228] LINEAR GROUP 325 
 
 of order p is permutable only with its own powers. There are 
 therefore two conjugate sets of operations of order p, each set 
 containing ^(p'^ — l) operations. Again, each of the operations 
 of a cyclical sub-group of order ^(p — l)or^(p+l)is conjugate 
 to its own inverse and to no other of its powers. Hence if 
 i (p + i) is even and therefore ^{p — 1) odd, there are i (p — 3) 
 conjugate sets of operations whose orders are factors of ^(p — 1), 
 each set containing p'+p operations; i(p~3) conjugate sets 
 of operations whose orders are factors of ^(p + 1), other than 
 the factor 2, each set containing p^ — p operations ; and a single 
 set of operations of order 2, containing \ {p^ — p) operations. If 
 ^ (^ — 1) is even and -^{p + l) odd, there are \(p — l) conjugate 
 sets of operations whose orders are factors of ^(p + 1), each 
 containing p'—p operations; i(p—5) conjugate sets whose 
 orders are factors of ^{p— 1), other than the factor 2, each 
 set containing p^ + p operations; and a single set of -^(p^+p) 
 conjugate operations of order 2. In either case, the group 
 contains, exclusive of identity, -^ (p + 3) conjugate sets of 
 operations. 
 
 228. Since p — 1 and p + 1 can have no common factor ex- 
 cept 2, it follows that, if q^ denote the highest power of an odd 
 prime, other than p, which divides the order of IT, q™ must be a 
 factor of ^(p — 1) or of ^(p + i); and the sub-groups of order 
 q™ must be cyclical. Moreover, since no two cyclical sub-groups 
 of order ^ (^ — 1), or ^ (jp -I- 1), have a common operation except 
 identity, the same must be true of the sub-groups of order g™. 
 
 If 2" is the highest power of 2 that divides ^(p — 1) or 
 ^(p + 1), 2™+' will be the highest power of 2 that divides the 
 order of H. Moreover, a sub-group of order 2™+^ must contain 
 a cyclical sub-group of order 2™ self-conjugately, and it must 
 contain an operation of order 2 that transforms every operation 
 of this cyclical sub-group into its own inverse ; in other words, 
 the sub-groups of order 2™+^ are of dihedral type. 
 
 Suppose now that two sub-groups of order 2™+' (m > 1) have 
 a common sub-group of order 2'' (r > 2). Such a sub-group 
 must be either cyclical or dihedral : in the latter case, it 
 contains self-conjugately a single cyclical sub-group of order 
 
326 THE FRACTIONAL [228 
 
 2''"'. Hence, on the supposition made, a cyclical sub-group of 
 order 4 at least would be contained self-conjugately in two 
 distinct cyclical sub-groups of order 2"*. It has been seen that 
 this is not the case ; and therefore the greatest sub-group, that 
 two sub-groups of order 2™+' can have in common, must be a 
 sub-group of order 4, whose operations, except identity, are all 
 of order 2. Now every groiip of order 2"'+^ contains one self- 
 conjugate operation of order 2, and 2™ operations of order 2 
 falling into 2 conjugate sets of 2^"^ each. Moreover, the group 
 of order p±l, which has a cyclical sub-group of order 2™ and 
 contains the operation A of order 2 of this cyclical group 
 self-conjugately, has ^{p ±1) other operations of order 2 ; and 
 
 p + 1 
 therefore it contains %^~^^ sub-groups of order 2™+', each of 
 
 which has A for its self-conjugate operation. If now B is 
 any operation of order 2 of this sub-group of order ^ + 1, and 
 if it is distinct from A, then B enters into a sub-group of 
 order 2™+^ that contains A self-conjugately. But since A is 
 permutable with B, A must belong to the sub-group of order 
 p±l, which contains B self-conjugately; hence A enters into 
 a sub-group of order 2™+^ which contains B self-conjugately. 
 The sub-group {A, B] is therefore common to two distinct 
 sub-groups of order 2"*+^ Now no group of order 2'' (r > 2) 
 can be common to two sub-groups of order 2™+^ ; and therefore 
 {A, B] must (§ 80) be permutable with some operation ;Sf whose 
 order s is prime to 2. If s is not 3, S must be permutable with 
 A and B: and then {A, S} and {B, S] would be two distinct 
 sub-groups of orders 2s, whose operations are permutable with 
 each other. It has been seen that H does not contain such 
 sub-groups. Hence s = 3 ; and S transforms A, B and AB cycli- 
 cally, or {8, A, B} is a sub-group of tetrahedral type (§ 202). 
 
 The number of quadratic* sub-groups contained in H may 
 be directly enumerated. A group of order 2'"+^ contains 2™^' 
 such sub-groups, which fall into 2 conjugate sets of 2'"~^ each; 
 a single group of order 8 containing each quadratic group self- 
 
 ffl + 1 
 
 conjugately. The quadratic groups, contained in the n^+^ 
 
 sub-groups of order 2*"+^ of a sub-group of order p ±1, are 
 * A non-cyclical group of order 4 is called a quadratic group. 
 
230] LINEAR GROUP 327 
 
 clearly all distinct, and each quadratic group belongs to just 3 
 groups of order ^ ± 1 ; thus {A, B} belongs to the 3 groups 
 which contain A, B and AB respectively as self-conjugate 
 operations. Hence the total number of quadratic groups con- 
 tained in H is 
 
 229. The greatest sub-group of a group of order 2"^^, that 
 contains a quadratic group self-conjugately, is a group of order 
 8 and dihedral type ; and it has been shewn that 3 is the 
 only factor, prime to 2, that occurs in the order of the sub- 
 group containing a quadratic group self-conjugately. Hence 
 finally, the order of the greatest group containing a quadratic 
 
 i»(«^ — 1) 
 group self-conjugately is 24, and the quadratic 
 
 Ip (if — 1) 
 groups fall into two conjugate sets of ^. each. The 
 
 group of order 24, that contains a quadratic group self-con- 
 jugately, contains also a self-conjugate tetrahedral sub-group, 
 while the sub-groups of order 8 are dihedral. Hence (§ 84) this 
 group must be of octohedral tjnpe. 
 
 Since every tetrahedral sub-group of H contains a quadratic 
 sub-group self-conjugately, and every octohedral sub-group 
 contains a tetrahedral sub-group self-conjugately, there must 
 also be two conjugate sets of tetrahedral sub-groups and two 
 conjugate sets of octohedral sub-groups, the number in each set 
 
 230. We have hitherto supposed m > 1, or what is the 
 "same thing, p=±\ (mod. 8). If now m = 1, so that p=±^ 
 (mod. 8), the highest power of 2 that divides the order of H is 
 2^; and, since 2^ is not a factor of ^{p ±1), the sub-groups of 
 order 2^ are quadratic. Moreover, since 2^ is the highest power 
 of 2 dividing the order of H, the quadratic sub-groups form 
 a single conjugate set. Each sub-group of order ^ + 1, which 
 has a self-conjugate operation of order 2, contains Up ±1) 
 
328 THE FRACTIONAL [230 
 
 sub-groups of order 4, and each of the latter belongs to 3 of the 
 former. The total number is as before 
 
 12 
 
 and since they form a single conjugate set, each quadratic group 
 is self-conjugate in a group of order 12. Also, for the same 
 reason as in the previous case, this sub-group is of tetrahedral 
 type. 
 
 Finally, since every sub-group of H of tetrahedral type 
 must contain a quadratic sub-group self-conjugately, H must 
 
 contain a single conjugate set of ^^ tetrahedral sub- 
 groups. In this case, the order of H is not divisible by 24, and 
 therefore the question of octohedral sub-groups does not arise. 
 
 231. The group H always contains tetrahedral sub-groups ; 
 when its order is divisible by 24, it contains also octohedral 
 sub-groups. Now ii p = ±l (mod. 5), the order of H is 
 divisible by 60 : it may be shewn as follows that, in these cases, 
 H contains sub-groups of icosahedral type. 
 
 Let us suppose, first, that p=l (mod. 5) ; and let j be a 
 primitive root of the congruence 
 
 f = 1 (mod. p). 
 
 Then ( tij) , which we will denote by J., is an operation of order 
 
 5. The operations of order 2 of H are all of the form B, where 
 
 B denotes ( ] , since each is its own inverse. Now 
 
 Kjx — aJ 
 
 AB = ('t±^\; 
 \yjx — aj V 
 
 and (§ 203) if A and B generate an icosahedral group, 
 
 (ABy=l. 
 
 A simple calculation shews that, if this condition is satisfied, 
 then 
 
 ciHj-j-J = i. 
 
 Also, since the determinant of B is unity, 
 
231] LINEAR GROUP 329 
 
 These two congruences have just p-1 distinct solutions, 
 the solutions a, ^, y and -a, -^, -7 being regarded as 
 identical. There are therefore p—1 operations of order two in 
 H, namely the operations 
 
 where ^, . _ l _ . . _^._^^^ , 
 
 which with A generate an icosahedral sub-group. 
 The group generated by 
 
 1 
 
 m - 
 
 ^« + /9o 
 
 yoX- 
 
 contains 5 of the p—1 operations of order 2 of the form 
 
 1 
 
 
 1 
 
 
 
 viz. those for which 
 
 ^ = /8„i", y = yoj^, (n = 0, 1, 2, 3, 4). 
 
 Hence the sub-group {A}, of order 5, belongs to ^(p — 1) 
 distinct icosahedral sub-groups. Now each icosahedral sub- 
 group has 6 sub-groups of order 5 ; and H contains ^p(p + l) 
 sub-groups of order 5 forming a single conjugate set. The 
 number of icosahedral sub-groups in H is therefore 
 
 The group of isomorphisms of the icosahedral group is the 
 symmetric group of degree 5 (§ 178). Now ZTcan contain no 
 sub-group simply isomorphic with the symmetric group of 
 degree 5. For if it contained such a sub-group, an operation of 
 
330 THE FRACTIONAL [231 
 
 order 5 would be conjugate to its own square ; and this is not 
 the case. 
 
 Hence (§ 165), if an icosahedral sub-group K oi IT is con- 
 tained self-conjugately in a greater sub-group L, then L must be 
 the direct product of K and some other sub-group. This also is 
 impossible ; for the greatest sub-group of H in which any cyclical 
 sub-group, except those of order p, is contained self-conjugately, 
 is of dihedral type. Hence L must coincide with K, and K 
 
 must be one of ^ 'I conjugate sub-groups. The icosa- 
 hedral sub-groups of IT therefore fall into two conjugate sets of 
 M|^ each. 
 
 In a similar manner, when p= — l (mod. 5), we may take as 
 a typical operation A, of order 5, 
 
 1,2-1 
 
 y — iP cc — iV 
 
 and it may be shewn, the calculation being rather more 
 cumbrous than in the previous case, that there are just p + 1 
 
 fcix + i3y 
 {ABy=i, 
 
 operations JB, of the form f j , such that 
 
 and that five of these belong to the icosahedral group gene- 
 rated by A and any one of them. It follows, exactly as 
 
 in the previous case, that H contains %f. icosahedral 
 
 sub-groups, which fall into two conjugate sets, each set con- 
 
 taming ^ • groups. 
 
 232. Finally, we proceed to shew that H has no other 
 sub-groups than those which have been already determined. 
 Suppose, first, that a sub-group h oi H contains two distinct 
 sub-groups of order p. These must, by Sylow's theorem, form 
 part of a set of kp + \ sub-groups of order p conjugate within 
 h. 'Now H contains only p + 1 sub-groups of order p, and 
 
232] LINEAR GROUP 331 
 
 therefore h must be unity and h must contain all the sub-groups 
 of order p ; or since H is simple, h must contain and therefore 
 coincide with H. Hence the only sub-groups of H, whose orders 
 are divisible by p, are those that contain a sub-group of order 
 p self-conjugately. They are of known types. 
 
 Suppose next that ^ is a sub-group of H, whose order n 
 is not divisible by p, and let S-^ be an operation of g whose 
 order q^ is not less than the order of any other operation of g. 
 In H the sub-group [Si] is self-conjugate in a dihedral group of 
 order p±\; and the greatest sub-group of this group, which 
 contains no operation of order greater than qi, is a dihedral 
 group of order 2qi. Hence in g the sub-group {Sj\ is self- 
 conjugate in a group of order q^ or iq^, and therefore it forms 
 
 one of — or of g— conjugate sub-groups. Moreover, no two of 
 
 these sub-groups contain a common operation except identity ; 
 
 and they therefore contain, excluding identity, — ^^ distinct 
 
 operations, where ei is either 1 or 2. 
 
 Of the remaining operations of g, let S2 be one whose 
 order q^ is not less than that of any of the others. The 
 
 Th ( O ^~ 1 ) 
 
 operation 82 cannot be permutable with any of the —^ 
 
 6l2'l 
 
 operations already accounted for, since S^ is not a power of any 
 one of these operations. Hence, exactly as before, [S^] must 
 
 form one of — conjugate sub-groups in g, e^ being either 1 or 
 e^qt 
 
 2 ; and these sub-groups contain -~- ^ operations which are 
 
 distinct from identity, from each other, and ■ from those of the 
 previous set. This process may be continued till the identical 
 operation only remains. Hence, finally, n being the total 
 number of operations of g, we must have 
 
 or - = 1 — 2, . 
 
 n V e^q^ 
 
332 THE FRACTIONAL [233 
 
 233. In this equation, let r of the e's be 1 and s of them be 
 2, so that r + s is their total number, say m. Then 
 
 < 1 - r + ^r - is + 4^ 
 
 < 1 - \m. 
 
 Hence, since w is a positive integer, there cannot be more 
 than three terms under the sign of summation. Moreover, since 
 
 r cannot be greater than 1, and therefore not more than one of 
 the e's can be unity. Also, when one of the e's is unity, we have 
 
 so that, in this case, s cannot be greater than unity. The 
 solutions are now easily obtained by trial. 
 
 (i) For one term in the sum, the only possible solution is 
 ei=l, w = g'i, 
 and the corresponding group is cyclical. 
 
 (ii) For two terms in the sum, the solutions are 
 
 («) e, = e,= 2,r.= -^; 
 
 2i ■•■ H^ 
 
 (/3) ei = 2, 6, = 1, (?, = 2, n=23i; 
 
 (7) 6, = 1, e, = 2, gi = 3, ^, = 2,n = 12. 
 
 To the solution (a) there corresponds no sub-group ; for 
 n< 2g'i, and the values qi = qi, ej = 2 imply that g has a sub- 
 group of order 2^1. 
 
 To the solution (y8) correspond the sub-groups of order 2qi 
 of dihedral type, for which q^ is odd, so that the operations of 
 order 2 form a single conjugate set. 
 
233] LINEAR GROUP 333 
 
 To the solution (7) corresponds a sub-group of order 12 
 containing 8 operations of order 3 and 3 operations of order 2, 
 i.e. a tetrahedral sub-group. 
 
 (iii) For three terms in the sum, the solutions are 
 
 (a) €, = 62 = 63 = 2, ^2=2, q, = 2, n = 2q,: 
 
 (B) „ „ ?i = 3, ^2 = 3, ^3 = 2, n = 12 
 
 (7) .. .. qi = 4, g'2 = 3, qs = 2, n = 24 
 
 (S) „ „ q, = 5, q,= ^, q, = 2, n = 60. 
 
 To the solution (a) correspond the sub-groups of order 2^, 
 of dihedral type, in which q^ is even, so that the operations of 
 order 2, which do not belong to the cyclical sub-group of order 
 5'i, fall into two distinct conjugate sets. 
 
 To the solution (/3) would correspond a group of order 12 
 containing 3 operations of order 2 and 4 sub-groups of order 8 
 which fall into two conjugate sets of 2 each. Sylow's theorem 
 shews that such a group cannot exist ; and therefore there is no 
 sub-group of H corresponding to this solution. 
 
 Solution (7) gives a group of order 24, with 3 conjugate 
 cyclical sub-groups of order 4, 4 conjugate cyclical sub-groups 
 of order 3, and 6 other operations of order 2 forming a single 
 conjugate set. No operation of this group is permutable with 
 each of the 4 sub-groups of order 3 ; and therefore, if the group 
 exists, it can be represented as a transitive group of 4 symbols. 
 On the other hand, the order of the symmetric group of 4 
 symbols, which (§ 203) is simply isomorphic with the octohedral 
 group, is 24; and its cyclical sub-groups are distributed as 
 above. Hence to this solution there correspond the octohedral 
 sub-groups of H. 
 
 Solution (S) gives a group of order 60, with 6 conjugate 
 sub-groups of order 5, 10 conjugate sub-groups of order 3, and 
 a conjugate set of 15 operations of order 2. It has been shewn, 
 in § 85, that there is only one type of group of order 60 that 
 has 6 sub-groups of order 5 ; viz. the alternating group of 
 degree 5 : and that, in this group, the distribution of sub-groups 
 in conjugate sets agrees with that just given. Moreover, the 
 alternating group of degree 5 is simply isomorphic with the 
 
334 GENERALIZATION OF [233 
 
 icosahedral group. Hence to this solution there correspond the 
 icosahedral sub-groups of H. 
 
 234. When ^ > 11, then \p (;? - 1) > 60 ; and, when p > 3, 
 \][>{p — \)>p + l. Hence when ^>11, the order of the 
 greatest sub-group of H is ^p (p — 1), and the least number of 
 symbols in which H can be expressed as a transitive substitution 
 group is p -I- 1 . 
 
 When ^ is 5, 7 or II, however, H can be expressed as a 
 transitive substitution group of p symbols^. 
 
 For, when p = 5, H contains a tetrahedral sub-group of 
 order 12, forming one of 5 conjugate sub-groups ; therefore H 
 can be expressed as a transitive group of 6 symbols. It is to be 
 noticed that in this case H is an icosahedral gi-oup. 
 
 When p=l, H contains an octohedral sub-group of order 
 24, which is one of 7 conjugate sub-groups ; and H can there- 
 fore be expressed as a transitive group of 7 symbols. Similarly, 
 when ^ = 11, iT contains an icosahedral sub-group of order 60, 
 which is one of 11 conjugate sub-groups; and the group can be 
 expressed transitively in 11 symbols. 
 
 235. The simple groups, of the class we have been dis- 
 cussing in the foregoing sections, are self-conjugate sub-groups 
 of the triply transitive groups of degree p -}- 1, defined by 
 
 the existence of which was demonstrated in § 113. In fact, 
 
 (ax + ^\ . (kax^k^\ . ,, 
 
 5 and -5 TtT represent the same trans- 
 
 \yx + bJ \kjx + kbj ^ 
 
 formation, the determinant, aS — I3y, of any transformation may 
 always be taken as either unity or a given non-residue ; and it 
 follows at once that the transformations of determinant unity 
 form a self-conjugate sub-group of the whole group of trans- 
 formations. 
 
 If, as in § 113, a, /3, 7, S, are powers of i, where i is a primi- 
 tive root of the congruence 
 
 *>"-! = 1, (mod. p), 
 
 1 This is another of the results stated in the letter of Galois referred to in 
 the footnote on p. 192. 
 
 Since 
 
235] THE FEACTIONAL LINEAR GROUP 335 
 
 the triply transitive group G of degree ^" + 1, which is defined 
 by the transformations, has again, when p is an odd prime, a 
 self-conjugate sub-group H of order \p^ (p^™ - 1), which is 
 given by the transformations of determinant unity. It follows 
 from Theorem IX, § 134, that G, being a triply transitive group 
 of degree ^" -f 1, must have, as a self-conjugate sub-group, a 
 doubly transitive simple group ; and it is easy to shew that H 
 is this sub-group. 
 
 In fact, if a simple group h is a, self- conjugate sub-group of 
 G it must be contained in H. Also, since A is a doubly 
 transitive gi'oup of degree p" 4- 1, it must contain every opera- 
 tion of order p that occurs in G. Now we may shew that these 
 
 operations generate H. Thus ( J and (« -f 2 — i — i~^) are 
 
 operations of order p belonging to G. Therefore ( ^^ 
 
 (ISC \ 
 -^j by 
 
 r ) . Hence ( ^^i ) belongs to h ; and a sub-group of /(, 
 
 which keeps one symbol unchanged is the group of order 
 
 -^—^ j . The order of h 
 
 therefore is not less than Jp" {p™ — 1) ; in other words h is 
 identical with H. 
 
 When p = 2, every power of i is a quadratic residue, and the 
 determinant of every transformation is unity. In this case it 
 may be shewn, by an argument similar to the above, that the 
 group G of order 2" (2^ - 1) is itself a simple group. 
 
 We are thus led to recognize the existence of a doubly- 
 infinite series of simple groups of orders 2™ (2^" - 1) and 
 ^p" (_p2m _ 1)^ which are closely analogous to the groups of order 
 \p (p' — 1) already discussed. For an independent proof of the 
 existence of these simple groups and for an investigation of 
 their properties, the reader is referred to the memoirs mentioned 
 below ^ 
 
 1 Moore: "On a doubly-infinite series of simple groups," Chicago Congress 
 Mathematical Papers, (1893) ; Burnside: "On a class of groups defined by con- 
 gruences," Proc. L. M. S. Vol. xxv, (1894), pp. 113—139. 
 
336 THE HOMOGENEOUS LINEAR GROUP [236 
 
 236. We will now return to the linear homogeneous group 
 G of transformations of n symbols, taken to a prime modulus p ; 
 and consider it more directly as the group of isomorphisms of 
 
 aa Abelian group of order p" and type (1, 1 to n units). 
 
 As in I 1.56, it may be expressed in the form of a substitution 
 group performed on the p" — 1 symbols of the operations, other 
 than identity, of the Abelian group. In this form it is clearly 
 transitive, since there are isomorphisms changing any' operation 
 of the Abelian group into any other operation. If P is any 
 operation of the Abelian group, an isomorphism which changes 
 any one of the p — \ operations 
 
 P,P\ PP-\ 
 
 into any other, will certainly interchange the set among them- 
 selves. Hence, when expressed as a group of degree p^—\, G 
 is imprimitive ; and the symbols forming an imprimitive system 
 are those of the operations, other than identity, of any sub- 
 group of order p of the Abelian group. If 
 
 are a set of generating operations of the Abelian group, an 
 isomorphism, which changes each of the sub-groups 
 
 {PiMP^j ,{Pn] 
 
 into itself, must be of the form 
 
 -'I ) "2 ) -t^n 
 
 P«i P oj P mi 
 
 1 > -^ 2 ) > -t^n . 
 
 )■ 
 
 «s 
 
 This isomorphism changes P1P2 into (PiPa"')"' ; therefore it 
 will only transform the sub-group {P1-P2} into itself when 
 «! s 02, (mod. p). If then the given isomorphism changes every 
 sub-group of order p into itself, we must have 
 
 0^ = 02= = an, (mod. p). 
 
 Hence the only operations of G, which interchange the 
 symbols of each imprimitive system among themselves, are 
 those given by the powers of 
 
 /-^i ) -^ 2 ) > -^ » \ 
 
 \P.^P.% ,PnV' 
 
 where a is a primitive root of p. This operation is the same as 
 
236] IN TRANSITIVE FORM 337 
 
 that denoted by -4 in § 217. It follows immediately that the 
 factor-group j^ can be represented as a transitive group in 
 
 »" — 1 
 _^ symbols. In fact, the operations of {A} are the only 
 
 operations of which transform each of the — =- sub-groups 
 
 «" — 1 
 of order p in itself; and these ~ r- sub-groups must be 
 
 permuted among themselves by every operation of G. The 
 substitution group thus obtained is doubly transitive ; for if P 
 and P' are any two operations of the Abelian group such that 
 P' is not a power of P, and if Q and Q' are any other two 
 operations of the Abelian group subject to the same condition, 
 there certainly exists an isomorphism of the form 
 
 P.P', 
 
 Q.Q'. 
 
 :)■ 
 
 and this isomorphism changes the sub-groups [P] and [P'} into 
 the sub-groups {Q} and [Q'}. 
 
 These results will still hold if, instead of considering Q the 
 total group of isomorphisms, we take T the group of iso- 
 morphisms of determinant unity. Thus the determinant of 
 
 (P'P'' ] 
 
 XQ^Q'. ; 
 
 is a. times the determinant of 
 
 \Q,Q', )' 
 
 It is therefore possible always to choose a so that the de- 
 terminant of 
 
 VQ",Q' ) 
 
 shall be unity; and this isomorphism still changes the sub- 
 groups {P} and {P'} into {Q] and {Q'} respectively. 
 
 p-i 
 The lowest power of A contained in T is (§ 217) -4 <* . 
 
 Hence the group -. — ^zy. can be represented as a doubly 
 
 {a^\ 
 
 22 
 
 B. 
 
338 SIMPLE GROUPS DEFINED BY [236 
 
 transitive group of degree *—-j . This group is (§ 220) simply 
 
 N . . 
 
 isomorphic with the simple group of order ^ — _.., , , which is 
 
 defined by the composition-series of G. 
 
 We may sum up these results as follows : — 
 
 Theorem. The homogeneous linear group of order 
 
 iV = (^" - 1) (p'^-p) (jJ™ -p"-^ 
 
 when p^ is neither 2^ nor 3^ defines, by its composition-series, a 
 simple group of order ryj > where d is the greatest common 
 
 factor of p—1 and n. This simple group can he represented as 
 a doubly transitive group of degree p^~^ + p"~^ + +^ + 1. 
 
 «"— 1 
 237. The - — — symbols, permuted by one of these doubly 
 
 transitive simple groups, may be regarded as the sub-groups of 
 
 order p of an Abelian group of order ^" and type (1, 1, to 
 
 n units). Now every pair of sub-groups of such an Abelian 
 
 group enters in one, and only in one, sub-group of order p^ ; 
 
 and every sub-group of order p^ contains p + 1 sub-groups of 
 
 »"— 1 
 order p. Hence from the =- symbols permuted by the 
 
 <(j™ — 1 M»-i — 1 
 doubly transitive group, z~—-^ — =— sets of p + 1 symbols 
 
 each may be formed, such that every pair of symbols occurs in 
 one set and no pair in more than one set, while the sets are 
 permuted transitively by the operations of the group. These 
 groups therefore belong to the class of groups referred to in 
 § 148. The sub-group, that leaves two symbols unchanged, 
 permutes the remaining symbols in two transitive systems of 
 
 p -1 and^"~^ +p'^~^ + +p^; and the sub-group, that leaves 
 
 unchanged each of the symbols of one of the sets oi p + 1, is 
 contained self-conjugately in a sub-group whose order is 
 (p + l)p times that of a sub-group leaving two symbols un- 
 changed. This latter sub-group permutes the symbols in two 
 
 transitive systems oi p + 1 and p'"~^ + p'^~^ + +p^. It may 
 
 be pointed out that, when n is 3, such a sub-group is simply 
 
238] THE HOMOGENEOUS LINEAR GROUP 339 
 
 isomorphic with, but is not conjugate to, the sub-groups that 
 leave one symbol unchanged: this may be seen at once by 
 noticing that an Abelian group, of order p^ and type (1, 1, 1), has 
 the same number of sub-groups of orders p and p^. 
 
 238. Some special cases may be noticed. First, when p = 2, 
 both p -1 and d are unity, and the homogeneous linear group is 
 itself a simple group. 
 
 If w = 3, then iV = 168 j so that the group of isomorphisms of a 
 group of order 8, whose operations are all of order 2, is the simple 
 group of order 168 (§ 146). 
 
 If n = 4, then iV = 2« . 3^ 5 . 7 . This is the order of the 
 alternating group of 8 symbols ; and it may be shewn that this 
 group is simply isomorphic with the group of isomorphisms. 
 
 The Abelian group of order 16 contains 35 sub-groups of order 
 4 ; and it may be shewn that, from these 35 sub-groups, sets of 5 can 
 be formed in 56 distinct ways, so that each set of 5 contains every 
 operation of order 2 of the Abelian group once, and once only. If 
 
 "l> "il "zi "i 
 
 are a set of generating operations of the Abelian group, such a set 
 of 5 sub-groups of order 4 is given by 
 
 [P., A}, {P„ A}, {P^P., P.P.], {P.P., P.P.P,], {P.P., PiP.P.}- 
 
 Now (^^' ^" ^^' ^' ) 
 
 is an isomorphism of order 7 of the Abelian group, and P^ is the 
 only operation of the group, except identity, which is left unchanged 
 by this isomorphism. It may be directly verified that the 7 sets of 
 5 groups of order 4, into which the given set is transformed by the 
 powers of this isomorphism, contain every sub-group of order 4 of 
 the Abelian group. The 7 sets, being interchanged among them- 
 selves by this isomorphism of order 7 which leaves only P-^ unchanged, 
 must be interchanged among themselves by isomorphisms of order 7 
 which leave any other single operation of the Abelian group 
 unchanged. There are therefore at least 15 isomorphisms of order 
 7 which interchange the 7 sets among themselves. Now the 
 isomorphisms, which interchange the 7 sets among themselves, form a 
 sub-group of the group of isomorphisms, which is isomorphic with a 
 group of degree 7 ; and the only groups of degree 7, which contain at 
 least 15 operations of order 7, are the symmetric and the alternating 
 groups. The group of isomorphisms must therefore contain a sub- 
 group which is isomorphic with the symmetric or with the alternating 
 group of degree 7. Hence at once, since the group of isomorphisms 
 is simple, it must contain a sub-group which is simply isomorphic 
 with the alternating group of degree 7. Since this must be one of 
 
 22—2 
 
340 GENERALIZATION OF [238 
 
 8 conjugate sub-groups, the group of substitutions itself is simply 
 isomorphic with the alternating group of degree 8. 
 
 If jo''=3», thenp''-'+ +^+1 = 13, d=l, and iV= 2^ 3M3. 
 
 There is therefore a doubly transitive simple group of degree 13 and 
 order 2^3^13 (§§145, 149). 
 
 239. The homogeneous linear group may be generalized by 
 taking for the coefficients powers of a primitive root of 
 
 ip'-~^ = l, (mod. ^), 
 
 instead of powers of a primitive root of 
 
 t^-i = l, (mod. p). 
 
 When the coefficients are thus chosen, the order of the group 
 Gj),n,^, defined by all sets of transformations 
 
 Xn ^ (IjiiXi + C^n2*^2 "T "T (^nn^n > 
 
 whose determinant differs from zero, may be shewn, as in § 172, 
 to be 
 
 N=(p"''-1) {p^'-p") {f"'-p"') (^"--pln-i)^); 
 
 and the order of the sub-group F, formed of those trans- 
 
 N 
 formations whose determinant is unity, is -;; — =• . The only 
 
 self-conjugate operations of V are the operations of the sub- 
 group generated by {ix^, ix^, , ixn), which are contained in 
 
 r. If 8 is the greatest common factor oi p"—! and n, these 
 self-conjugate operations of T form a cyclical sub-group 7 of 
 order S. Finally, the argument of § 219 may be repeated to 
 
 r 
 
 shew that - is a simple group. 
 
 The homogeneous linear group Op^n,v> when values of v 
 greater than unity are admitted, thus defines a triply infinite 
 system of simple groups ; it may be proved that these groups 
 can, for all values of v, be expressed as doubly transitive groups 
 
 puv _ J 
 
 of degree *- =- , 
 
240] THE HOMOGENEOUS LINEAR GROUP 341 
 
 240. We may shew, in conclusion, that the group G^p,„,„ is 
 simply isomorphic with a sub-group of (r^,„„,i. For this purpose, 
 we consider the group defined by 
 
 cc,' = asi + i*-., a;,' =x, + i'-2, , x^^ ^x^ + {•■«, 
 
 (n, »-2, »-»=o, 1, 2, , r-1); 
 
 the congruences being taken to modulus p. This is an Abelian group 
 of order p"" and type (1, 1, to mv units). Moreover, the opera- 
 tion 
 
 Xj = ajiSJi -I- + cti,ja;„, 
 
 ^n — ^n\ *a + + (^nn^n > 
 
 of ^P,n,v transforms the given operation of the Abelian group into 
 where i*i = «iii''i + a^^i''^ + + a^^i''", 
 
 %"« = a^,i'^< + a„„i'» -I- -I- a„„i" 
 
 Every operation of Gj,^n,v, as defined in § 239, is therefore 
 permutable with the Abelian group, and gives a distinct isomorphism 
 of it; or in other words, as stated above, G'^j.^^.-is simply isomorphic 
 with a sub-group of Gp,nv,i- 
 
 Further, the sub-group 
 
 *^ =^ Xi + 1 f X^ = ^2 ) J '^11 -— ^n 1 
 
 (r = o,i,2, ,r-i). 
 
 is transformed by the given operation of Gp^,^^„ into the sub-group 
 
 (r = 0, 1,2, ,r-l)- 
 
 If a^ = a.ii= =«„i = 0, 
 
 the two sub-groups are identical; but if these conditions are not 
 satisfied, they have no operation in common except identity. 
 Moreover, 
 
 may each have any value from to iP"'^, simultaneous zero values 
 alone excluded. Hence the sub-group of order p" defined by 
 
 X^ := flT^ -f- t , X2 = *^2 ) , n ^— '^ll f 
 
 {r = 0, 1, ,r-l). 
 
 is one of r- conjugate sub-groups in the group formed by com- 
 bining the Abelian group with (?j,,„,„ ; and no two of these conjugate 
 sub-groups have a common operation except identity. 
 
342 EXAMPLES [240 
 
 The p"" - 1 operations, other than identity, of an Abelian group 
 of order p"" and type (1, 1, to nv units), can therefore be 
 
 divided into ^ r sets of «" — 1 each, such that each set, with 
 
 p" -I 
 identity, forms a sub-group of order p" ; and the group ff^^„,„ is iso- 
 morphic with a group of isomorphisms of the Abelian group which per- 
 
 »"" — 1 
 mutes among themselves such a set of ^ — ^ sub-groups of order p". 
 
 Ex. 1. Shew that the^^ ^-^-^ — = — ~- sub-groups of 
 
 p-\ .p'—V /)"— 1 ° "^ 
 
 order p" of an Abelian group of order ;?'"■ and type (1, 1, to 
 
 p"v _ \ 
 nv units) can be divided into sets of — — tt- each, such that each set 
 
 contains every operation of the group, other than identity, once and 
 once only ; and discuss in how many distinct ways such a division 
 may be carried out. 
 
 Ex. 2. Shew that the simple group, defined by the group of 
 isomorphisms of an Abelian group of order />" and type (1, 1, ..., 1), 
 admits a class of contragredient isomorphisms, which change the 
 operations of the simple group, that correspond to isomorphisms 
 leaving a sub-group of order p of the Abelian group unaltered, into 
 operations that correspond to isomorphisms leaving a sub-group of 
 order p^~'^ of the Abelian group unaltered. 
 
CHAPTER XV. 
 
 ON SOLUBLE AND COMPOSITE GROUPS. 
 
 241. The most general problem of pure group-theory, so 
 far as it is concerned -with groups of finite order, is the deter- 
 mination and analysis of all distinct types of group whose order 
 is a given integer. The solution of this problem clearly involves 
 the previous determination of all types of simple groups whose 
 orders are factors of the given integer. 
 
 Now there is no known criterion by means of which we can 
 say whether, corresponding to an arbitrarily given composite 
 integer N as order, there exists a simple group or not. For 
 certain particular forms of N, this question can be answered in 
 the affirmative. For instance, when N = ^n\, previous in- 
 vestigation enables us to state that there is a simple group of 
 order N; but even in these cases we cannot, in general, say 
 how many distinct types of simple group of order N there are. 
 
 On the other hand, we have seen that, for certain forms of N, 
 it is possible to state that there is no simple group of order JSf. 
 Thus, when N is the power of a prime, or is the product of two 
 distinct primes, there is no simple group whose order is N; and 
 further, every group of order N is soluble. Again, if JV" is 
 divisible by a prime p, and contains no factor of the form 
 1 + kp, a group of order N cannot be simple ; for, by Sylow's 
 theorem, it must contain a self-conjugate sub-group having a 
 power of j5 for its order. 
 
 We propose, in the present Chapter, to prove a series of 
 theorems which enable us to state, in a considerable variety of 
 
344 GROUPS WHOSE ORDERS CONTAIN [241 
 
 cases, that a group, which has a number of given form for its 
 order, is either soluble or composite. If the results appear 
 fragmentary, it must be remembered that this branch of the 
 subject has only recently received attention : it should be 
 regarded rather as a promising field of investigation than as 
 one which is thoroughly explored. 
 
 The symbols Pi, P2, Ps, will be used throughout the 
 
 Chapter to denote distinct primes in ascending order of magni- 
 tude ; while distinct primes, without regard to their magnitude, 
 will be denoted by ^, g', r*, 
 
 242. Theorem I. 1/ H is one of n conjugate sub-groups 
 of a group G of order N, and if N is not a factor of n\, G 
 cannot he simple. 
 
 The group G is isomorphic with the group of substitutions 
 given on transforming the set of n conjugate sub-groups by 
 each of the operations of G. This is a transitive group of 
 degree n, and its order therefore is equal to or is a factor of n !. 
 Hence, if iVis not a factor of n\, the isomorphism cannot be 
 simple ; G is therefore not simple. 
 
 Corollary*. If a prime p divides N, and if m is the 
 greatest factor of N which is congruent to unity (mod. p), G 
 will be composite unless iV is a factor of m !. 
 
 There cannot, in fact, be more than m conjugate sub-groups 
 whose order is the highest power of p that divides N ; so that 
 this result follows from the theorem. 
 
 In dealing with a given integer as order, it may happen that, 
 though no single application of this theorem will prove the corre- 
 sponding group to be composite, a repeated application of Sylow's 
 theorem, on which the theorem depends, will lead to that result. 
 As an example, let us consider groups whose order is 3^ . 5 . 11. Any 
 such group must contain either 1 or 45 sub-groups of order 11, and 
 either 1 or 1 1 sub-groups of order 5. Hence, if the group is simple, 
 it must contain 45 sub-groups of order 11, and 11 sub-groups of 
 order 5 ; and each one of the latter must be contained self-conju- 
 gately in a sub-group of order 45. Now the 45 sub-groups of order 
 11 would contain 450 distinct operations of order 11, leaving only 
 45 others; and therefore a sub-group of order 45, if the group contain 
 such a sub-group, must in this case be self-conjugate. This is, 
 
 * Holder, Math. Ann. Vol. xl, (1892), p. 57. 
 
243] NOT MORE THAN TWO DISTINCT PRIMES 3i5 
 
 however, in direct contradiction to the assumption that the group 
 contains 1 1 sub-groups of order 5 ; therefore the group cannot be 
 simple. 
 
 243. Theorem II. Every group whose order is the power 
 of a prime is soluble. 
 
 This has already been proved iu § 54. 
 
 Theorem III*. A group G whose order is p^q^, where a is 
 less than 2m, m being the index to which p belongs (mod. q), is 
 soluble. 
 
 li a<m, G contains a self-conjugate sub-group H of order 
 ?"• 
 
 If a = m, G contains either 1 or p™ sub-groups of order q^. 
 In the latter case, if q^ is the order of the greatest sub-group 
 common to two groups of order q^, and if h is such a sub- 
 group, h must (§ 80) be contained self-conjugately in a 
 sub-group k of order p''q''+^, x>0, s > ; and this sub-group 
 must contain more than one sub-group of order 5''+'. Hence x 
 must be m; and therefore h must be common to all the p™ 
 sub-groups of order q^, so that h is self-conjugate, and s is 
 jS — r. If r is zero, no two sub-groups of order q^ have a 
 common operation except identity, and the ^™ sub-groups 
 contain p"^(q^ — l) distinct operations, so that a sub-group 
 of order p™ is self-conjugate. Hence when a is equal to m, G 
 contains either a self-conjugate sub-group of order q^ (r > 0) or 
 else one of order p"'- 
 
 If m < a < 2m, G contains 1 or p™ sub-groups of order q^. 
 If there is only one, it is self-conjugate. 
 
 If there are p™ sub-groups of order q^, and if q^ (r > 0) is 
 the order of the greatest group common to any two of them, 
 it may be shewn, exactly as in the preceding case, that G 
 contains a self-conjugate sub-group of order q''. 
 
 If r = 0, the q^ — 1 operations, other than identity, of each 
 of the ^'" sub-groups of order q^ are all distinct. Now G is 
 isomorphic with a group of degree p™. If the isomorphism 
 is multiple, G must have a self-conjugate sub-group whose order 
 is a power of p. If the isomorphism is simple, we may regard 
 
 * Frobenius, Berliner Sitzangsberichte 1895, p. 190, and Burnside, Proc. 
 L. M. S. Vol. XXVI, (1895), p. 209 ; for the case where a-i^m. 
 
346 GROUPS WHOSE ORDERS CONTAIN [243 
 
 G^ as a transitive group of degree p™; and every operation, 
 except identity, of a sub-group of order q^ will leave one symbol 
 only unchanged. Hence q^ is equal to or is a factor of p*" — 1. 
 Moreover, when G is thus regarded, the sub-groups of order p" 
 must be transitive; for all the sub-groups of order q^ will be 
 given on transforming any one of them by the operations of a 
 sub-group of order p". 
 
 Consider now a sub-group of order p^-^^(f, which keeps one 
 symbol fixed and contains a sub-group of order (f self-conju- 
 gately. It will contain 5''(7</8) sub-groups of order ■p'--^'^. 
 Any one of these must keep p" symbols fixed ; for it is the 
 sub-group of a transitive group of order p° and degree p^, which 
 keeps one symbol fixed. If 7 < /3, there are operations of order 
 q which transform one of these sub-groups into itself, and 
 therefore interchange among themselves the p^ symbols un- 
 changed by the sub-group. Hence p'—Y must be divisible 
 by q. This requires that x=m, which is impossible. Hence 
 7 = /3, and the sub-group of order p'^^'^q^ contains q^ sub-groups 
 of order p'^'^. It is obvious that no two of these can enter in 
 the same sub-group of order p" ; for if they did, they would 
 generate a group of order p'^-^'^+o' (a; > 0), and a sub-group 
 of this order cannot enter in a group of order p'^''^q^. Therefore 
 G contains q^ sub-groups of order j)" 
 
 Since the sub-groups of 0, of order p"^, are transitive, their 
 self-conjugate operations must (§ 106) displace all the symbols, 
 and they cannot therefore be permutable with any operation 
 whose order is a power of q. Suppose now that every 
 operation of a sub-group Q, of order q^, is permutable with 
 an operation P of order p", and that there is no sub-group 
 of order 5*+^ of which this is true. If /S is a self-conjugate 
 operation of a sub-group of order p" to which P belongs, 
 P is transformed into itself and Q into a new sub-group 
 Q' by 8. Hence the sub-group which contains P self-con- 
 jugately has two, and therefore p™, sub-groups of order <f. 
 No two of these can belong to the same sub-group of order 
 q^, for if the}'' did they would generate a group of order 
 qx+x' ^gf > 0). Now since P is permutable with Q, it must 
 leave unchanged the symbol which Q leaves unchanged. 
 Hence P must leave every symbol unchanged ; i.e. it must 
 
243] NOT MORE THAN TWO DISTINCT PRIMES 347 
 
 be the identical operation. The order of every operation of Q 
 is therefore either a power of p or a power of q. 
 
 Suppose now that p" is the order of the greatest sub-group 
 that is common to any two sub-groups of order p", and that h 
 is such a sub-group. Then (§ 80) h must be permutable with 
 an operation of order q. Since no operation of h is permutable 
 with any operation of order q, it follows that ^"^ — 1 is divisible 
 by q, and therefore that r = m. 
 
 Let k, of order p^+^qy, be the greatest sub-group that 
 contains h self-conjugately. If 7 = /?, h is common to each 
 of the q^ sub-groups of order p", and it is therefore a self- 
 conjugate sub-group of G. 
 
 If 7</3, let Q be a sub-group, of order q"*, of k. In 
 there must be a group, whose order is divisible by 5^+^ 
 containing Q self-conjugately. Hence there must be operations 
 in G, which transform Q into itself and h into a conjugate 
 sub-group h'. If h and /(,' have a common sub-group, it must 
 be transformed into itself by every operation of Q. This is 
 impossible, since p* — 1 is not divisible by q when s<m. 
 
 In the group {h', Q], Q is one of p™ conjugate sub-groups, 
 and therefore no operation of h' can transform Q into itself. 
 Hence if an operation P' of h' transforms h into itself, it must 
 transform Q into a group Q', which is conjugate to Q in k. 
 Hence, k' being the greatest sub-group that contains h' 
 self-conjugately, {Q, Q'} must be common to k and k'. But 
 [Q> Q'}, containing two sub-groups of order qy, must contain p™ 
 such sub-groups ; and therefore the p^ sub-groups of order qy 
 that enter in k are identical with those that enter in k'. 
 This is impossible if H and H' are distinct ; for the p™ sub- 
 groups of k, of order qy, generate {h, Q}. Hence no operation 
 of h', except identity, transforms h into itself, and the 
 number of sub-groups in the conjugate set to which h belongs 
 must not be less than p"^ ; so that 
 
 pw. ^^a-in^S^p-y. 
 
 Now we have seen that 
 
 p™=l (mod. q^), 
 and q^ = 1 (mod. ^''-'"■) ; 
 
SiS GROUPS WHOSE ORDERS CONTAIN [243 
 
 from these congruences, and the inequality a — m<in, it follows 
 
 that 
 
 pm ^ pa-mqP _ q? + l_ 
 
 This inequality is inconsistent with the previous one, which 
 follows directly from the supposition that 7 is less than j8. 
 Hence 7 = /3, and A is a self-conjugate sub-group of G. 
 
 It follows therefore that, on every possible supposition, G 
 must have a self-conjugate sub-group. If this sub-group be 
 represented by Gi, the same reasoning may be repeated with 
 
 respect to the groups ^ and Gi. Hence G is soluble. 
 
 Corollary. All groups whose orders are pi/^a", Pi'Pi^, 'Px'Pi', 
 PiPi, PiP-2°; Pi'Pi are soluble. 
 Since the congruence 
 
 p^ = 1 (mod. p^ 
 
 is satisfied only by p^ = 2, p2 = 3, the results stated follow 
 immediately from the theorem, except for the cases 2^3" and 
 2^3". 
 
 If in these cases there are 16 sub-groups of order 3" (a > 1), 
 there must (§ 78) be sub-groups of order 3""^ common to two 
 sub-groups of order 3"; and, in the groups of order 2^3" or 
 2'3°, such a sub-group, if not self-conjugate, must be one of 
 either 4 or 8 conjugate sub-groups. The groups must therefore 
 be isomorphic with groups of degree 4 or 8 ; from this it follows 
 immediately that they must be soluble (§ 146). 
 
 244. Theorem IV. Groups of order p-^pi are soluble. 
 
 If a group G, of order p^^p^, contains only p^ sub-groups of 
 order jJi", it cannot be simple, for a group of order p^ cannot be 
 expressed as a substitution-group of p2 symbols. Similarly, if G 
 contains a single group of order p^°-, it is not simple. 
 
 If G contains pi sub-groups of order p^', and if these sub- 
 groups have no common operations, except identity, there are 
 P-ZiPi' — 1) operations in G whose orders are powers of pi] and 
 therefore a group of order p^^ is self-conjugate. If the operations 
 of the p2^ sub-groups of order pi' are not all distinct, let jp/ be 
 the order of the greatest sub-group common to any two of them ; 
 
244] NOT MORE THAN TWO DISTINCT PRIMES 349 
 
 and let h be such a sub-group. Then (§ 80) h must be self- 
 conjugate in a group k of order p-i'+^pi (s > 0, /3 = 1 or 2). If 
 ^ = 2, A, is self-conjugate in G ; and if ^=\,k must contain p^ 
 sub-groups of order p-i!'*\ and therefore 
 
 ^2 = 1 (mod. ^i)- 
 
 Now (§ 78) pi' = 1 (mod. Pi'-% 
 
 and therefore ^2=1 (mod. Pi""''), 
 
 unless pi = 2; but, if jOj = 2, then 
 
 ^2 = + 1 (mod. pi"-^-^). 
 
 Again, ii s—a — r, h is one of ^2 conjugate sub-groups; as 
 before, G cannot then be simple. 
 
 Suppose now that H is a, sub-group of order p^", and that 
 the self-conjugate operations of H form a sub-group h. This 
 must be one of 1, ^2 or pi conjugate sub-groups ; in the first 
 two cases, G cannot be simple. Moreover, if any two sub-groups 
 of the conjugate set to which h belongs have a common 
 sub-group, it must be self-conjugate in a group containing 
 more than one sub-group of order pi»; again, G cannot be 
 simple. 
 
 Let H, Hi, , -Hj,j_i 
 
 be the p2 sub-groups (§ 80) of order pi" that contain h ; and 
 suppose that h is not contained in h. Every operation of each 
 of the Pi sub-groups 
 
 11) 111 ) 11^2—1 > 
 
 is permutable with every operation of h. But these sub-groups, 
 since they occur in k and not in h, generate a sub-group of k, 
 whose order is divisible by p^. Hence there must be an operation 
 Q, of order p^, which is permutable with every operation of h. 
 This operation would be permutable with every operation of a 
 group of order pi'pi at least, and it would therefore be one of 
 Pi""'' conjugate operations at most. Now, if^i>2,^i°~^<^2; and 
 as a group of order pi cannot be represented in terms of less 
 than p^ symbols, G would in this case have a self-conjugate 
 sub-group. If pi = 2, then either pi^~''< 2^2. or Pi''^ =2(p2+l). 
 A group of order pi cannot be expressed in less than 2^2 
 symbols ; a cyclical group of order pi cannot be expressed in 
 
350 GROUPS WHOSE ORDERS CONTAIN [244 
 
 2(^2 + 1) symbols; and a group of degree 2(^2 + 1), which 
 contains a non-cyclical sub-group of order pi, must (§ 141) 
 contain the alternating group. Hence again, in this case, G 
 must have a self-conjugate sub-group. 
 
 Finally, suppose that A contains the ^2 sub-groups 
 
 h, hi, , h^^_i. 
 
 If h, contains further sub-groups of the conjugate set to 
 which h belongs, they must be permuted among themselves in 
 sets of ^2 when A is transformed by an operation Q, of order -p^, 
 belonging to h. Hence we may assume that 
 
 h, hj , , n^j— 1 , 
 
 "l)!' %2+i ! napa— 1) 
 
 n (a;— 1)^)2) n (a;— 1)^)2+1 ) > "■X'Pi-\t 
 
 is the complete set of sub-groups of the conjugate set to which 
 h belongs, contained in A; and that, when transformed by Q, 
 those in each line are permuted among themselves. If {Q, Q'} 
 is a sub-group* of order p^, the pi sub-groups 
 
 b, hi, , hp^2_i, 
 
 are permuted transitively among themselves, when transformed 
 by the operations of {$, Q'}. Hence if Q' change h into hj,^,^, it 
 must change the set 
 
 b, hi, hi>2-i. 
 
 into the set 
 
 "^VVi. > "j/Pi+l ' J " {y+i)p2-i j 
 
 and therefore 
 
 is a set of ^2 sub-groups of order p,^, which have the common 
 sub-group Q'~%Q'. 
 
 Now, since by supposition A is not a self-conjugate sub- 
 group of H, there must (§ 55) be in ^ a sub-group A' conjugate 
 to and permutable with A. Let this be the sub-group of order 
 Pi", common to 
 
 H, H'l, , H'p^_i. 
 
 No one of the sets 
 
 ^yp^,' ^yp2+i' ' ^(y+i)P2-i, (2/ = 0, 1. «- 1), 
 
 * If this sub-group is cyclical, then Q = Q''. 
 
245] NOT MORE THAN TWO DISTINCT PRIMES 351 
 
 can have more than one group in common with this set, as 
 otherwise p-^^ would not be the order of the greatest group 
 common to two groups of order p^"^. Hence, of the set of groups 
 
 h, hi', , h'j,^_i, 
 
 at least p^ — x are distinct from the groups, conjugate to h, 
 which enter in h. Every group of order p^'^" occurring in k will 
 have a similar set oi p^ — x sub-groups of the set h, which do 
 not occur in h. Hence, since the operations of h are the only 
 operations common to two sub-groups of order pi"^' in k, there 
 must be in k not less than p^ (p. —x) + p^, i.e. pi, sub-groups 
 of the set to which h belongs. 
 
 In other words, k must contain a self-conjugate sub-group 
 of G. 
 
 On every supposition that can be made, we have shewn that 
 
 must have a self-conjugate sub-group. If this sub-group is 
 
 n 
 (?i, and if the order of either Gi or ^ contains p^, we may 
 
 n 
 repeat the same reasoning ; while if the orders of ^ and Gi are 
 
 both divisible by p2, we have already seen that they must be 
 soluble. Finally, then, G itself must be soluble. 
 
 245. Theorem V. A group of order pi-p2^, in which the 
 sub-groups of orders fi- and p^^ are Abelian, is soluble. 
 
 Suppose, first, that there are p^y (7 < /3) sub-groups of order 
 Pi". The group is then isomorphic with a substitution group 
 of degree p2''. Now the operations of a sub-group of order p/ 
 transform the p^y sub-groups of order p^"- transitively among 
 themselves ; and therefore, in the isomorphism between the 
 given group and the substitution group of degree pa'*'. there 
 corresponds to an Abelian sub-group of order p/ a transitive 
 substitution group of degree p^y. But (§ 124) an Abelian group 
 can only be represented as a transitive substitution group of a 
 number of symbols equal to its order. Hence the isomorphism 
 between the given group and the substitution group is multiple ; 
 and the given group is not simple. 
 
 Suppose, next, that there are ^2^ sub-groups of order p^", and 
 therefore (§ 81) pi" — 1 distinct sets of conjugate operations 
 
352 GROUPS WHOSE SUB-GROUPS OF ORDER p^ [245 
 
 whose ordei's are powers of pi. Let P be an operation whose 
 order is a power of ^i; and let H, of order 'px"P^> be the greatest 
 group which contains P self-conjugately. Then H contains at 
 least |>2* operations whose orders are powers of pa, and therefore 
 at least 'p} distinct operations of the form PQ, where P and Q 
 are permutable and 
 
 Q*'^* = 1. 
 
 Now P is one of ff'^ conjugate operations; corresponding 
 to each of these, there is a similar set of p/ operations of the 
 form PQ ; while no two such operations can be identical. The 
 group therefore contains 'pf operations of the form PQ : and 
 similarly, it contains an equal number for each set of conjugate 
 operations whose order is a power of f^. Hence, finally, the 
 group contains p^ (pi" — 1) operations whose orders are divisible 
 by pi. There is therefore in this case a self-conjugate sub-group 
 of order pa- 
 
 The group therefore always contains a self-conjugate sub- 
 group. A repetition of the same reasoning shews that it is 
 soluble. 
 
 246. Theorem VI. A group of order pi'-^pi^ Pii^, in 
 
 which the sub-groups of order Pt!^\ p^ P^-i ^"""^ cyclical, is 
 
 soluble *. 
 
 If the number of operations of the group, whose orders divide 
 
 Pr'^'p'^l Pn"", is exactly equal to this number, it follows 
 
 from Theorem VII, Cor. I, § 87, that the same is true for the 
 number of operations of the group whose orders divide 
 
 p°^j p„"" . Now, when r = 1 , this relation obviously holds ; 
 
 and therefore it is true for all values of r. The group therefore 
 contains just pn" operations whose orders divide p»""; hence 
 a sub-group of order jJn"" is self-conjugate. Any sub-group of 
 order p^ll must transform this self-conjugate sub-group of 
 order p^"" into itself, so that the group must contain a sub- 
 group of order p^rjpn""- If there were more than one sub- 
 group of this order, the group would contain more than 
 P»-i^"°" operations whose orders divide this number. Hence 
 * Proc. L. M. S. Vol. xxvi, (1895), p. 199. 
 
247] ARE ALL CYCLICAL 353 
 
 the group contains a single sub-group of order p^'ZlPn"^" > ^■nd 
 this sub-group must be self-conjugate. By continuing this 
 reasoning it may be shewn that, for each value of r, the group 
 
 contains a single sub-group of order Pr'"p°^'^l Pn"", such a 
 
 sub-group being necessarily self-conjugate. The group is there- 
 fore soluble; for we have already seen that a group whose 
 order is the power of a prime is always soluble. 
 
 The self-conjugate sub-group of order p/' pn"- must 
 
 contain the complete set of sub-groups of order p^". If 
 then there are m of these sub-groups, m must be a factor of 
 
 p'"^J Pn"-" ; and a sub-group of order pr'^ must be contained 
 
 self-conjugately in a sub-group of order 
 
 K' P: 
 
 m 
 
 Hence if i and j are any two indices between 1 and n, the group 
 contains sub-groups of order pi'^pf. Now the above sub-group, 
 which contains a sub-group of order p^"^ self-conjugately, is of 
 the same nature as the original group. Hence, i and j being 
 any two indices less than r, it contains a sub-group of order 
 Pi^Vj^p/'- This process may be continued to shew that, if fi 
 is any factor of ^i"' pn""" which is relatively prime to 
 
 ^' ' Pn" ^ |.jjg^ ^jjg group contains sub-groups of order /t. 
 
 /^ 
 Ex. Shew that, when the groups of order p^"^ are also cyclical, 
 there are sub-groups whose orders are any factors whatever of the 
 order of the group. 
 
 247. A special case of the class of groups under consideration 
 is that in which the order contains no repeated prime factor. These 
 groups have formed the subject of a memoir by Herr O. Holder'. 
 He shews that they are capable of a specially simple form of repre- 
 sentation, which we shall now consider. 
 
 Let N' = piPi Pn 
 
 be the order of a group G. It has been seen above that a sub-group 
 of order p„ is self -conjugate. Suppose then that pi, Pm^ , Fn ^^re 
 
 1 " Die Gruppen mit quadratfreier Ordnungszahl," GSttingen Nachrichten, 
 1895 : pp. 211—229. Compare also Frobenius, Berliner Sitzungsberichte, 1895, 
 pp. 1043, 1044. 
 
 23 
 
354 THEOREMS GIVING [247 
 
 the orders of those cyclical sub-groups of prime order which are 
 contained self-conjugately in G. They generate a self-conjugate 
 
 sub-group H of order pip,^ p„, say /a. Each of the cyclical 
 
 sub-groups [Pi], {P^}, , {Pn) of H, generated by an operation of 
 
 prime order, is permutable with all the rest ; and therefore (§ 34) 
 
 the operations Pi, P^, -?» are all permutable. Hence, since 
 
 their orders are distinct primes, the sub-group H is cyclical. Let 
 now X be a sub-group of G of order v, where iV = /xi/. No self- 
 conjugate sub-group of G can be contained in K. For if it contained 
 such a sub-group of order p^pp Ft (" < /^ < < y), this sub- 
 group would contain a single cyclical sub-group of order py-, and G 
 would contain a self-conjugate sub-group of order Py, contrary to 
 supposition. Since then K contains no self-conjugate sub-group of 
 G, it follows (§ 123) that G can be expressed transitively in /* 
 symbols ; when it is so expressed, an operation M of order /*, that 
 generates H, will be a circular substitution of fi symbols. The 
 only substitutions performed on the /«. symbols, which are per- 
 mutable with 3f, are its own powers; and therefore no operation of 
 G is permutable with M except its own powers. Every operation 
 of K must therefore give a distinct isomorphism of {M}. Now 
 (§ 169) the group of isomorphisms of a cyclical group is Abelian. 
 Hence K must be Abelian ; and since its order contains no repeated 
 prime factor, it must be cyclical. The group G is therefore com- 
 pletely defined by the relations 
 
 M>^=1, iV^- = 1, N-'MN = M% 
 
 where a belongs to index v, mod. fn. 
 
 248. The theorem of § 246 is a particular case of the 
 following more general one, due to Herr Frobenius^ from which 
 a number of interesting and important results may be deduced. 
 
 Theorem VII. If is a group, of order N = mn, where m 
 and n are relatively prime, and where 
 
 m=p'^q^ ry ; 
 
 and if sub-groups P, Q, , R, of orders p'^,<f, , r'^, are 
 
 N N 
 
 -, e(Q) and — 
 
 Abelian, while 6{P) and — , 0{Q) and -^-^ , 6{R) and 
 
 N . . 
 
 — , are relatively prime ; then contains (i) exactly n operations 
 
 whose orders divide n, and (ii) a sub-group of order m which 
 has a self-conjugate sub-group of order r^. 
 
 The theorem will be proved inductively, by shewing that, if 
 1 " tjber auflosbare Gruppen, II " -. Berliner Sitaungsherichte, 1895, p. 1035. 
 
248] FURTHER TESTS OF SOLUBILITY 355 
 
 it is true for any similar group of order m'n', where m' is a 
 factor of m, it is also true for G. 
 
 Suppose that the greatest group, which contains i? self- 
 
 conjugately, is a group H' of order p'^'q^' r^n', where n' is a 
 
 factor of n. Let N' be any operation of this sub-group, whose 
 order v' is a factor of n'. Then (§ 177) since 6 (R) and v are 
 relatively prime, N' is permutable with every operation of R. 
 Hence every operation of H', whose order divides n', is per- 
 mutable with every operation of R. 
 
 Let now S be any operation of R ; and suppose that the 
 order of K, the greatest sub-group which contains 8 self- 
 conjugately, is p^'q^' rv??i. If the order of 8 is r^, the 
 
 order of ^m is p^'q^^ r^-^, say min-i. We have seen in 
 
 § 176 that, if Pj is a sub-group of P, then 9 (Pi) is equal to or 
 
 is a factor of (P). Hence if Pj is a sub-group of order p"' of 
 
 K JSf 
 
 rm> 9 (Pi) and — are relatively prime; therefore, a fortiori, 
 
 6 (Pi) and -^-^ are relatively prime. Similarly 0(Qi) and 
 
 ' are relatively prime, and so on. Moreover mi is a factor 
 
 of m. We may therefore assume that the theorem holds for 
 
 j-^ . Hence this group contains exactly Wi operations whose 
 
 orders divide %, and it also contains a sub-group of order 
 Ml in which a sub-group of order rv~* is self-conjugate. There- 
 fore K contains a sub-group of order p'^^q^^ r> in which a 
 
 sub-group of order rT is self-conjugate, and exactly Wj operations 
 of the form 8N, where 8 and N are permutable and the order 
 of N divides n. 
 
 Now 8 is one of p'^-'-iqP-P' — conjugate operations in 
 
 G; corresponding to each of these, there is such a set of Wi 
 operations of the form 8N, while (§ 16) no two of these 
 
 operations can be identical. Hence G contains p^-^ig^s-^, ^ 
 
 operations of the form 8N, arising from the conjugate set to 
 which *Sf belongs. If then we sum for the distinct conjugate 
 sets of operations of G whose orders are powers of r, the 
 
 23—2 
 
356 THEOREMS GIVING [248 
 
 number of operations of G, whose orders divide r^n and do not 
 divide n, is n%p°-~°-^q^~^^ 
 
 Now since H' is the greatest group that contains R self- 
 
 conjugately, the greatest common sub-group of K and H' is 
 
 the greatest sub-group of K that contains It self-conjugately. 
 
 The order of this group certainly contains p'^^q^^ r'v as 
 
 a factor ; and if the order is ^"i q^' r^i/, then 8 is one of 
 
 n' . 
 pa'-a, g,3'-Pi _ conjugate operations in H'. Every operation 
 
 of H' however, whose order divides n', is permutable with every 
 operation of R ; and therefore v = n'. Hence the number of 
 operations of H', exchiding identity, whose orders are powers of r, 
 
 is I'p'^'-'^iq^'-^' , where the summation is extended to all the 
 
 distinct conjugate sets of operations of H' whose orders are 
 powers of r. It has been shewn (Theorem IV. Corollary, § 81) 
 that the number of distinct conjugate sets of operations in G, 
 whose orders are powers of r, is the same as the number in H'. 
 Hence the symbols X and 2' represent summations of the same 
 number of terms. 
 
 Finally, in accordance with the induction we are using, we 
 may assume that the number of operations of G whose orders 
 divide r-vra is exactly equal to ri'm. For the order of G may be 
 separated into the factors 
 
 m' =p'^q^ > 
 
 and n' = rTn, 
 
 where m' is a factor of m, while the conditions of the theorem 
 hold for this separation. 
 
 Hence the number of operations of G whose orders divide ?i, 
 is equal to 
 
 n (rT — Xp'^-'^iqP-Pi ), 
 
 while at the same time 
 
 rT - 1 = Sp"'""'/'"^' 
 
 Unless a' = a, ^' = ^, the former of these numbers is 
 
 negative, which is impossible ; these conditions must there- 
 fore be satisfied. The number of operations, whose orders 
 divide n, is therefore exactly equal to n ; and the order of 
 the greatest group H' that contains R self-conjugately is 
 
248] FURTHER TESTS OF SOLUBILITY 367 
 
 Tf' 
 p^q^ rfn. The factor-group ^ has for its order ^"g^ n, 
 
 and it satisfies the same conditions as 0. Hence it contains a 
 sub-group of order p'^q^ ; and H' therefore contains a sub- 
 group of order p'^q^ ■j-t. 
 
 The free use of the inductive process that has been made in the 
 preceding proof may possibly lead the reader to doubt its validity. 
 He will find it instructive to verify the truth of the theorem 
 directly in the simpler cases. In fact, the direct verification for 
 the case in which m is a power of a prime is essential to the formal 
 proof ; it has been omitted for the sake of brevity. 
 
 Corollary I. If m = rrvjm^, where mi and m^ are relatively 
 prime, G has a single conjugate set of sub-groups of order mj. 
 
 That contains groups of order mj may be shewn in- 
 ductively at once. For since it is clearly true when m contains 
 only one, or two, distinct prime factors, we may assume it true 
 when mi does not contain rf as a factor. Now G has a sub- 
 group g of order p'^q^ r-v, which contains a sub-group R of 
 
 order r^ self-conjugately. The factor-group -^ contains then a 
 
 IfYh 
 
 sub-group of order m/, where m,' is any factor of — which is 
 
 relatively prime to — ;— ; and therefore g contains a sub-group 
 of order miry. 
 
 If now / and I' are two sub-groups of G of order m^r'', 
 they may be assumed to contain the same sub-group R of 
 order r''. For the sub-groups of G of order rv form a single 
 conjugate set; and therefore, if 1' contained a sub-group R' of 
 this set, 8-^I'S would contain S'^R'S, which may be taken 
 to be R. Now R is a, self-conjugate sub-group of both 
 
 / /' 
 
 / and /', and therefore the factor-groups -^ and ^ of order m/ 
 
 W 
 
 are sub-groups of the factor-group --^ . Hence if the result is 
 
 true when mi does not contain the factor r^, it is also true 
 when TOi does contain r^. But when mj is equal to p% the 
 result is obviously true ; and therefore it is true generally. 
 
358 THEOREMS GIVING [248 
 
 Corollary II. If the sub-groups P, Q , R oi G 
 
 contain characteristic sub-groups Fq, Qo, , -Bo of orders 
 
 p"^, (fo, ,r'*''; then G contains a sub-group of order 
 
 m^=p'^q^<' r"/", which has a self-conjugate sub-group of 
 
 order ri"". 
 
 Assuming the truth of this statement when «j„ does not 
 contain the factor r, the factor-group -p- must contain a sub- 
 group of order — ; and therefore G contains a sub-group / of 
 
 order p'"og'^» r'^, which has R for a self-conjugate sub-group. 
 
 Hence, since R^ is a characteristic sub-group of R, J contains a 
 
 sub-group of order p'^'q^' and a self-conjugate sub-group 
 
 of order r"^'. It therefore contains a sub-group of order 
 p'^q^' 7^', which has a self-conjugate sub-group of order r^». 
 
 249. The reader will have no difficulty in seeing, as has 
 already been stated, that Theorem VI is a direct result of the 
 theorem proved in the last paragraph. We shall proceed at 
 once to further applications of it. 
 
 Theorem VIII. A gy-oup G of order 
 N = Pj'P2'^ p,^«», 
 
 171 which the suh-groups of every order p,.'^' {r = 1, 2, ,m — 1) 
 
 are Ahelian, with either one or two generating operations, is 
 generally soluble ; the special case pi = 2, pa = 3, may constitute 
 an exception^. 
 
 If an Abelian sub-group P,. of order p/^ is generated by a 
 single operation, it is cyclical and 6 (P^) is (§176) equal to p^ — 1. 
 If Pr is generated by two independent and permutable opera- 
 tions, (Pr) is equal to (p^ — 1) (^/ — !)• Now no prime greater 
 than Pr can divide (pr — 1) (p,-^ — 1) unless ^^ -I- 1 be a prime ; 
 and this is only possible when p^ is equal to 2. Hence unless 
 
 Pi = 2, Pa = 3, 6 (Pr) and — ^ are relatively prime for 
 
 all values of r from 1 to m — 1. Omitting for the present this 
 exceptional case, the conditions of Theorem VII are satisfied 
 by G; it therefore contains exactly p„'^' operations whose 
 
 1 Frobenius, loe. cit., p. 1041. 
 
249] FURTHER TESTS OF SOLUBILITY 359 
 
 orders are divisible by pn- Therefore G has a self-conjugate 
 sub-group Pn of order ^„"», and hence also one or more sub- 
 groups of order p^'lpn"^"- There are however only p^'llPn^" 
 operations in Q whose orders are divisible by no prime smaller 
 than pn-i, and therefore the sub-group of order p'^l^lpn'^" must 
 be self-conjugate. This process clearly may be continued to 
 shew that Q has a self-conjugate sub-group of every order 
 Pr'^P^i Pn"' (»' = 2, 3, n); from which it follows im- 
 mediately that Q is soluble. 
 
 Returning now to the exceptional case, suppose that a group 
 
 of order 2"' is contained self-conjugately in a maximum group 
 
 of order 1°-^2?m. If every operation of the group of order 2"' is 
 
 self-conjugate within this sub-group, Q contains 2"i — 1 distinct 
 
 conjugate sets of operations whose orders are powers of 2 ; 
 
 N 
 hence it follows that there are just ^ operations in Gt whose 
 
 orders are not divisible by 2. In this case, the conditions of 
 Theorem VII are satisfied by G ; and it is still soluble. 
 
 If, lastly, the operations of the sub-groups of order 2"i3^»i, 
 whose orders are powers of 2, are not all self-conjugate, there 
 must be an operation B, whose order is a power of 3, in this 
 sub-group which is not permutable with every operation of 
 the sub-group of order 2"'. Let now 
 
 be a characteristic series of the group A of order 2"' ; and 
 
 suppose that A^+i is the greatest of these groups with every 
 
 one of whose operations B is permutable. Then (§ 175) B is 
 
 A A 
 
 not permutable with every operation of -j-^ . Hence -j-^ 
 
 is a quadratic group ; and its three operations of order 2 must 
 be permuted cyclically when transformed by B. It follows 
 that {Ar, B\ is multiply isomorphic with a tetrahedral group. 
 Hence finally, under the conditions of the theorem, G is 
 certainly soluble unless it contains a sub-group which is 
 isomorphic with a tetrahedral group. 
 
 Corollary. A group of order p^^p^"" ...pn", in which no 
 one of the indices aj, a^, ..., a„_i is greater than 2, is soluble 
 
360 CERTAIN COMPOSITE GROUPS [249 
 
 unless it has a sub-group which is isomorphic with a tetrahedral 
 group. For a group of order i^r is necessarily an Abelian group 
 which is generated by either one or two independent operations. 
 
 250. We shall next consider certain groups of even order 
 in which the operations of odd order form a self-conjugate sub- 
 group. If (t is a group of order N, where 
 
 N = 2''«, (n odd), 
 we shall suppose, in this and the following paragraphs, that a 
 sub-group 2 of order 2" is Abelian, and that Q(2) and n are 
 relatively prime. Let us take first the case in which 2 is 
 cyclical, so that ^(^) is unity and the latter condition is satisfied 
 for all values of n. If such a group is represented in regular 
 form as a substitution group of I'^n letters, the substitution 
 corresponding to an operation of order 2" will consist of n cycles 
 of 2' symbols each. This is an odd substitution ; and therefore 
 (r has a self-conjugate sub-group of order 2"~%. In this sub- 
 group there are cyclical operations of order 2""^ Hence the 
 same reasoning will apply to it, and it contains a self-conjugate 
 sub-group of order t'-^n. Now (Theorem VII, Cor. i. § 87) G 
 contains exactly 2"~^?i operations whose orders are not divisible 
 by 2""^ Hence the sub-group of order '2.'^~''n must be self- 
 conjugate in Q. Proceeding thus, it may be shewn that G 
 contains self-conjugate (and characteristic) sub-groups of every 
 order I'-'^n (r-=l, 2, , a)*. 
 
 251. Suppose, secondly, that, under the same conditions, G 
 contains a self-conjugate sub-group ^ of order 2". Since 6{^2) 
 and n are relatively prime, every operation of 2 is permutable 
 with every operation of G' (§ 177). If 2' is a sub-group of 2 
 of order 2""^ 2' must therefore be self-conjugate in G. Now 
 
 the order of the factor group -^, is 2?i ; hence, by the preceding 
 
 result, it has a self-conjugate sub-group of order n. It follows 
 that G has a self-conjugate sub-group of order 2''~'m. This 
 group, again, has a self-conjugate sub-group of order 2''~°?i, 
 and so on. Hence G has a sub-group of order n. Now by 
 Theorem VII (§ 248), G has just n operations whose orders 
 
 * Frobenius, "tjber auflosbare Griippen, II" Berliner SiUungsberichte, 
 1895, p. 1039. 
 
252] OF EVEN ORDER 361 
 
 divide n. Hence the sub-group H, of order n, must be self- 
 conjugate; and G is the direct product of the two groups 3 
 and H. 
 
 252. Suppose next that 2 is not self-conjugate in ; and 
 let /, of order 2"^', be the greatest group that contains 3 self- 
 conjugately. Every operation of T is permutable with every 
 operation of S; and therefore (§ 81) G contains 2"~' distinct 
 sets of conjugate operations whose orders are powers of 2. 
 The case, in which 2 is cyclical, has been already dealt with 
 and will now be excluded ; we may thus assume that 3 contains 
 2^ — 1 (/8 -^ 2) operations of order 2, and that G contains an equal 
 number of sets of conjugate operations of order 2. 
 
 If possible, let no two sub-groups of order 2" have a common 
 operation except identity. Then if two operations S and T, 
 of order 2, are chosen, belonging to distinct conjugate sets and 
 to different sub-groups of order 2", they cannot be permutable 
 with each other. They will therefore generate a dihedral sub- 
 group of order 2m. If m were odd, 8 and T would be conjugate 
 operations. Hence m must be even ; and the dihedral sub-group 
 must contain a self-conjugate operation U of order 2. Since 
 U is permutable with both 8 and T, it must occur in at least 
 two different sub-groups of order 2"; and therefore the sup- 
 position, that no two sub-groups of order 2" have a common 
 operation except identity, is impossible. 
 
 Let now 2' of order 2"' be a sub-group common to 3 and 3^, 
 two sub-groups of order 2" ; and suppose that no two sub-groups 
 of order 2" have a common sub-group which contains 3' and is 
 of greater order. Since every operation of 3 is self-conjugate 
 in /, ^' must be self-conjugate in /. If then /, of order 2''/iV, 
 is the greatest sub-group of G that contains S' self-conjugately, 
 J contains r sub-groups of order 2\ Hence the factor-group 
 
 ^, contains r sub-groups of order 2""""', and no two of these 
 
 have a common sub-group ; for if they had, some two sub-groups 
 of order 2" contained in / would have a common sub-group 
 greater than and containing %', contrary to supposition. The 
 
 r sub-groups of order 2"-"' of ^ must therefore be cyclical. 
 
362 CERTAIN COMPOSITE GROUPS [253 
 
 253. We will apply the results of the last paragraph to 
 
 the case in which all the operations of 2 are of order 2. Then 
 
 2 
 "2' must be of order 2°'~^ since ^ is cyclical. Suppose now that 
 
 R is an operation of J of odd order, which is permutable with 
 :f but not with 2. If R were self-conjugate in a sub-group 
 whose order is divisible by 2", this sub-group would contain 
 :f and therefore one or more sub-groups of order 2" containing 
 ,'v'. But i2 is not permutable with any sub-group of order 2" 
 that contains £' ; and therefore the highest power of 2 that 
 divides the order of the group in which R is self-conjugate is 
 2°-~', so that R is one of 2/a (/a odd) conjugate operations. If 
 now A is an operation of order 2 that belongs to 2 and not to 
 'Ji', no operation conjugate to R can be permutable with A. 
 Hence in tlie substitution group of degree 2/j,, that results on 
 transforming the set of operations conjugate to R among them- 
 selves by all the operations of Q, the substitution corresponding 
 to A is an odd substitution. This substitution group has 
 therefore a self-conjugate sub-group whose order is half its own, 
 and therefore Q has a self-conjugate sub-group of order 2'^~^n. 
 In the same way it may be shewn that this sub-group has a 
 self-conjugate sub-group of order 2°'~^«.; and so on. Hence 
 has a sub-group H of order n. But it follows from Theorem 
 VII, § 248, that G has exactly n operations whose orders divide 
 n; and therefore H is a, self-conjugate sub-group. Moreover, 
 
 /-r 
 
 since jf is an Abelian group of order 2", it must contain self- 
 conjugate sub-groups of every order 2''(r=l, 2, , a— 1); 
 
 and G therefore has self-conjugate sub-groups of every order 
 2''n. In general however these sub-groups are not character- 
 istic, as is the case when '£ is cyclical. 
 
 254. We will consider next the case where 2 is generated 
 by two operations A and B, of orders 2""^ and 2. There are in 
 ',3 three operations of order 2, namely, ^^° , J.'^" B, and B ; and 
 in £ there are no operations, of order greater than 2, of which 
 the two latter are powers. Suppose that G contains an opera- 
 tion B', conjugate to B, with which A^'^~' is not permutable. 
 Then {J.^" , B'} is a dihedral group, and if R is an operation 
 
255] OF EVEN ORDER 363 
 
 of odd order of this group, no power of A is permutable with 
 
 jR. Since A^ and B' are not conjugate operations in G, there 
 
 must be an operation of order 2, conjugate to J.^" B, and 
 permutable with B. The sub-group, in which [B] is self- 
 conjugate, therefore contains representatives of each of the 
 three sets of conjugate operatioos of order 2 that belong to G; 
 and in this sub-group these representatives form three distinct 
 
 conjugate sets. Hence no operation conjugate to J.^" can be 
 permutable with B; and therefore no power, except identity, 
 of an operation conjugate to A is permutable with B. Hence 
 B is one of 2"~^/i. {/j, odd) conjugate operations, with none of 
 which is A permutable. It follows that, in the substitution 
 group of degree 2""^ with which G is isomorphic, A is an odd 
 substitution. Hence G contains a self-conjugate sub-group of 
 order 2'^~^n; and since this is of the same type as G, the 
 same reasoning may be applied to it. Exactly as before, G 
 will contain self-conjugate sub-groups of every order 2'"k 
 
 (r = 0, 1, , a-1). 
 
 We have assumed that there is an operation B', conjugate 
 
 to B, with which J.^" is not permutable. If A^ were per- 
 mutable with every operation that is conjugate to B, the 
 
 sub-group in which A^ is permutable would contain a self- 
 conjugate sub-group G' of G, whose order is divisible by 2. In 
 
 this case, we may deal with -^, exactly as we have been 
 
 dealing with G. 
 
 The results of §§ 250 — 254 may be summed up in the 
 following form: — 
 
 Theorem IX. If in a group G of order 2"^, where n is 
 odd, a sub-group 2 of order 2" is an Ahelian group of type 
 
 (a), (a— 1, 1), or (1, 1, , 1), and if 6{2) and n are relatively 
 
 prime, then G contains self -conjugate sub-groups of each order 
 2^n(0 = O,l, , a-1). 
 
 255. Still representing the order of G by 2»?i, where n is 
 odd, there are two cases, in which the sub-groups of order 2» 
 are not Abelian, where it may be shewn without difficulty that 
 G contains a self-conjugate sub-group of order n. 
 
364 CERTAIN COMPOSITE GROUPS [255 
 
 Theorem X. If the order of is 2"^, where n is odd and 
 not divisible by 3, and if a sub-group 2 of order 2" is of the type 
 
 (i) ^=""' = 1, £^ = J.2°~', B-^AB = A-\ 
 
 or (ii) ^^"~' = 1, 5^=1 , B-^AB = A-\ 
 
 Q contains a self -conjugate sub-group of each order 2^n 
 (/3 = 0, !,...,«- 1). 
 
 For each of these types, ^ (^) is 3 or 1 ; and if ^' is any 
 sub-group of S, ^ {3') is either 3 or 1. Hence if any operation 
 of G of odd order is permutable with ^ or £', it must be 
 perrautable with every operation of S or ^'. In each type, 
 the self-conjugate operations form a cyclical sub-group of order 
 2, namely {A^'^~^}. 
 
 We will consider first the case where 2 is of type (i). In 
 this case, [A^" ] and {B} are sub-groups of G of order 4, the 
 former being self-conjugate in 2 while the latter is not. If 
 they are conjugate sub-groups in G, we have seen (§ 82) 
 that G must contain an operation of odd order S, such that 
 the sub-groups S~^{B} S'^{n = 0, 1, 2,...) are permutable with 
 each other. This is impossible, since every operation that 
 is permutable with a sub-group ^' is permutable with all its 
 
 operations. Hence {A^"' } and {B} are not conjugate sub- 
 groups; and A^ and B are not conjugate operations. 
 
 Now A^° is self-conjugate in [A] and is one of two conjugate 
 operations in 2; therefore in G it must be one of 2/j, con- 
 jugate operations, where ft, is odd. When these 2/i conjugate 
 operations are transformed by A""^ , two only, namely, J."" and 
 A"^'' , remain unchanged. Let us suppose that, in the resulting 
 substitution, there are x cycles of 2 symbols and y cycles of 4 
 symbols. 
 
 When the 2/i conjugate operations are transformed by B, 
 none can remain unchanged; and we may suppose that the 
 resulting substitution has x' cycles of 2 symbols and y' cycles of 
 4 symbols. 
 
 Now since A^''~^=B\ 
 
 and therefore 1 +x = x'. 
 
256] OF EVEN ORDER 365 
 
 Hence one of the two substitutions J.^""' and B must be odd ; 
 G has therefore a self-conjugate sub-group of order 2''-^n. The 
 groups of order 2"~^ contained in this self-conjugate sub-group 
 are either of type (i), with a — 1 written for a : or they are 
 cyclical ; for, like 2, they can only contain a single operation of 
 order 2. If they are cyclical, the reasoning of § 250 may 
 be applied ; and if they are of type (i) the same reasoning 
 will apply to the self-conjugate sub-group of order 2'^-'^n 
 that has been used for G. Finally, then, G must contain 
 sub-groups of orders n, 2n, 2%..., 2"~'n, each of which is 
 contained self-conjugately in the next; and therefore the sub- 
 group H of order n must be self-conjugate. 
 
 If ^ is of type (ii), A is one of 2//, conjugate operations 
 in G, where fi is odd. It may be shewn, as in the previous 
 
 case, that B and A^ cannot be conjugate in G, so that no one 
 of the operations conjugate to A has B for one of its powers. 
 If now B were permutable with any one of these 2fi conjugate 
 operations, the group would contain an Abelian sub-group of 
 order 2", which is not the fact. Hence the substitution, given 
 on transforming the 2fi operations by B, consists of fi transposi- 
 tions and is an odd substitution. Therefore G contains a self- 
 conjugate sub-group of order 2''~^n, in which the sub-groups of 
 order 2""' are cyclical. Hence, again, there is a self-conjugate 
 sub-group of order n. 
 
 256. The only non- Abelian groups of order 2' are those of 
 types (i) and (ii) of § 265, when a = 3. The Abelian groups 
 of order 2^ and 2' are all included in the types considered in 
 Theorem IX, § 254. For an Abelian group ^, of order 2^ and 
 type (2, 1), 0(2) is 3 ; and for one of order 2' and type (1, 1, 1), 
 6(2) is 21. Hence Theorems IX and X shew that, if 2, 
 2= or 2= divide the order of a group, but not 2\ then the 
 operations of odd order form a self-conjugate sub-group, with 
 possible exceptions when 12 or 56 is a factor of the order. 
 Hence : — 
 
 Theorem XI. A group of even order cannot be simple 
 unless 12, 16 or 56 is a factor of the order *- 
 
 * Burnside, Proc. L. M. S. Vol. xxvi, (1895), p. 332. 
 
366 GROUPS WHOSE ORDERS CONTAIN [257 
 
 257. In further illustration of the methods of the preceding 
 paragraphs, we will deal with another case in which the sub-groups 
 of order p" are not Abelian. 
 
 Let (t be a group of order jo"m, where tn is relatively prime to 
 
 •p{ji — \) (^«— 1); and suppose that a sub-group ZTof order jB» is 
 
 such that within it every operation is either self -conjugate or one 
 of ^ conjugate operations*. Let A be a sub-group of G whose order 
 is a power of p ; and let S be an operation of 6 whose order is not 
 
 divisible by p. Then since m is relatively prime to p(jp-V) 
 
 {j)'^—\), if S is permutable with A, it is permutable with every 
 operation of A. Hence it follows from § 82 that no operation of 
 H, which is not self-conjugate in H, can be conjugate in G to 
 a self-conjugate operation of H. Suppose now that *Si and S^ are 
 two operations of H, each of which in U is one of <p conjugate 
 operations ; and that while j^i and S^ are conjugate in G, they 
 are not conjugate in /, the greatest group that contains S self- 
 conjugately. Let H-^ and H^ be the sub-groups of H, of orders 
 23""', which contain S-^ and S^ self-conjugately. Since S-^ and S^ 
 are conjugate in G, all the sub-groups of order ^''~' which contain 
 either of them self-conjugately belong to the same conjugate 
 set. If then H^ and H^ are identical, there must be an operation, 
 of order prime to p, which will transform 5^^ into S^ and H-^ 
 into itself. This, we have seen, is impossible. If H^ and H^ are 
 not identical, there must be an operation which will transform S-i 
 into -S'2 and H-^ into H^. Now H^ and H^\>QAh contain the sub-group 
 h formed of the self-conjugate operations of H ; and since no self- 
 conjugate operation of H is conjugate in G with any operation of 
 // which is not self -conjugate, the operation in question, which 
 transforms S-^ into S^ and H^ into H^, must transform h into itself. 
 Hence S-^ and S^ are conjugate operations in that sub-group, G' , of G 
 
 G' 
 which contains h self-conjugately. Now in -^ the operations, that 
 
 ZT 
 
 correspond to S-^ and S^, are self- conjugate operations of -r- . Since 
 then these operations are conjugate in -^, they must (§ 82) be 
 conjugate in r- . This however is impossible, since every operation 
 
 of r- whose order is a power of p is self-conjugate. Finally, then, no 
 
 two operations of -ff, which are not conjugate in H, can be conjugate 
 in G ; and the number of distinct sets of conjugate operations in G, 
 whose orders are powers of p, is equal to the number in H. From this 
 
 * The reader will easily verify that, when this condition is satisfied, t- 
 is Abelian, h being the sub-group formed by the self-conjugate operations of II, 
 
2-'58] NOT MORE THAN FIVE PRIME FACTORS 367 
 
 it follows, as in previous cases, that the number of operations of G 
 whose orders divide m, is equal to m. 
 
 258. Theorem XII. The only simple groups, whose orders 
 are the products of four or of five primes, are groups of orders 
 60, 168, 660, and 1092 : and no group, whose order contains 
 less than four prime factors, is simple. 
 
 Groups, whose orders are Pi,piP'i,pi,pi^, Pi'p2,piP2^ ovp^p^ps, 
 are all proved to be soluble by previous theorems in the present 
 chapter. 
 
 If the order of a group contains four prime factors, it must 
 be of one of the forms p^*, p^^p.,, pfp^, p^p^p^, p^pi, pipips, 
 P1P2P3 or PiPiPsPi- If Pi is an odd prime, groups of any one of 
 these orders have already been shewn to be soluble; while if jOj 
 is 2, the only case which can give a simple group is 2^p^pg. If 
 a group of this order is simple, it follows from Theorem VIII, 
 Cor. (§249), that p^ must be 3 ; and the order of the group 12^3. 
 A cyclical sub-group of order ^3 must be one of l+kp^ conjugate 
 sub-groups. Hence 1 -I- kps must be a factor of 12, so that ps is 
 either 5 or 11. If ps were 11, the 12 conjugate sub-groups of 
 order 11 would contain 120 distinct operations of order 11, and 
 the tetrahedral sub-group, which the group (if simple) must 
 contain, would be self-conjugate. Hence p^ must be 5 and the 
 order of the group is 60. We have already seen that a simple 
 group of order 60 actually exists, namely, the icosahedral group ; 
 and that there is only one type for such a group (§ 85). 
 
 If the order of a group contains five prime factors, it must 
 be of one of the forms : — 
 
 Pi, PlPi, P^P^, Pl%P3, PlPi, PlPiP%,PlPiPi, PlP^PsPi, PlP2\ 
 
 PiPips, PxP^pi, PiPip^Pi, PiPiPi, PiP^PsPi, PiP^P^Pi, PiP^PsPiPs- 
 
 lipi is an odd prime, it follows from previous theorems that 
 none of these forms, except pip^Ps sioA p^p^p^, can give simple 
 groups. 
 
 Taking first the order p^p^p^, let us suppose (without limita- 
 tion to the particular case) that a group of order p^p^p^ is 
 simple. It must contain p^ or p^p^ conjugate sub-groups of 
 order p™. In either case, the operations of these sub-groups 
 must be all distinct, or else they must all have a common 
 
368 GROUPS WHOSE ORDERS CONTAIN [258 
 
 sub-group, for the order of the group contains no factor 
 congruent to unity (mod. p^, except p^ or p^pz. Now the group 
 contains (§ 248) just pa^Ps operations whose orders are not 
 divisible by p^ ; and if there are ps sub-groups of order p^ 
 whose operations are all distinct, there are {p^ — \)ps operations 
 whose orders are powers oip^. Hence, in this case, a sub-group 
 of order p^ is self-conjugate. On the other hand, if there are 
 PiP^ sub-groups of order p^, their operations cannot be all 
 distinct ; and the group has a self-conjugate sub-group whose 
 order is a power of ^2. A repetition of this reasoning shews 
 that a group of order p^p^pz is always soluble. 
 
 We consider, next, a group whose order is p^p^p^. If p^ is 
 odd, p^ cannot be congruent to unity, mod. p^ or mod. p,. The 
 same is true, when p^ = 2, if p^ is not equal to 3. We shall 
 therefore first deal with a group of order pip^ps on the sup- 
 position that P2 is not 3. 
 
 If neither p^ nor p, divides p^' — 1, it follows from §§ 248, 257 
 that there are just piPs operations in the group whose orders 
 divide P2P3. When this is the case, the group clearly cannot 
 be simple. 
 
 If ps divides ^/— 1, and if there are more than psPs 
 operations whose orders divide p^pz, a sub-group of order p/ 
 must be self-conjugate in a sub-group of order jj/pj, so that the 
 group is isomorphic with a group of degree p^. In this case, 
 again, the group cannot be simple. 
 
 Finally, then, we have only to deal with the case in which a 
 sub-group of order p^' is self-conjugate in a sub-group of order 
 Pi%, while in this sub-group an operation of order p^ is 
 one of pi' conjugate operations. Now the congruences 
 
 Pj^= 1 (mod-^s), 
 p, = l{mod.pi), 
 
 are inconsistent ; therefore, if a sub-group of order p^ is self- 
 conjugate in a sub-group of order pi^Pi, the latter must be 
 Abelian. Hence the group contains Pi{p2 — ^)P3 operations 
 whose orders are divisible by p^. Now if the sub-groups of 
 order p^' have a common sub-group, it must be common to all of 
 the sub-groups of order pi^, and it is a self-conjugate sub-group. 
 
258] NOT MORE THAN FIVE PRIME FACTORS 369 
 
 If, however, no two have a common sub-group, the group 
 contains {pi~V)Pi operations of order p^; and there remain 
 only Pa operations whose orders divide p^. The group is 
 therefore in any case composite. 
 
 If now ^2 = 3, a group of order 2^*3^33 has, by Sylow's theorem, 
 a self-conjugate sub-group of order p^ unless p.^ is 5, 7, 11 or 23. 
 l^Pz = 23, the group (if simple) would have just 24 operations 
 whose orders divide 24;; it is thence easily seen to be non- 
 existent. If ^ = 11, the group (if simple) could be expressed as 
 a doubly transitive group of degree 12. In this form, however, 
 each operation, which transforms an operation of order 11 into 
 its own inverse, would be a product of 5 transpositions and 
 therefore an odd substitution. This group therefore cannot be 
 simple. If ps = 5, the group could be expressed as a doubly 
 transitive gi'oup of degree 6. The sub-group of order 4 which 
 transforms a sub-group of order 5 into itself would be cyclical ; 
 and the corresponding substitution being odd, the group could 
 not be simple. 
 
 Hence, finally, the only possibility is a group of order 168. 
 That a simple group of this order actually exists is shewn in 
 § 146 ; also, there is only one type of such group. 
 
 When px is equal to 2, the only case, that requires dis- 
 cussion in addition to those we have dealt with, is p^PiPiP^. 
 A group of this order can only be simple (§ 249) when it 
 contains a tetrahedral sub-group. In this case, the operations 
 of order 2 form a single conjugate set, and the group contains 
 just ^PiPi operations whose orders are divisible by 2. If a sub- 
 group of order 2'"3g' is the greatest that contains a sub-group of 
 order 3 self-conjugately, the group will contain 22-"(3 - \)piPi 
 operations whose orders are divisible by 8 and not by 2. Hence 
 the group must contain either p^pi, bp^Pi, or Ip^pi, operations 
 whose orders divide p^pi. On the other hand, the number of 
 these operations may be expressed in the form 
 «'(^3-l)^4 + 2/(P4-l)+l> 
 where a; is a factor of 12, and y is, & factor of 12^3 which is 
 congruent to unity, mod. 1)4. 
 
 Hence 
 
 a'(Pa-l)l'4 + 2/(P4-l) = ^i'3i'4-l, (^=1, 5, 7). 
 R. 24 
 
370 SIMPLE GROUPS WHOSE ORDERS [258 
 
 The case z=l leads to ?/ = 1, so that the sub-group of order 
 Pi is self-conjugate. 
 
 The case z= 5 gives no solution ; but when = 7, it will be 
 found that there are two solutions, namely, 
 
 x=6, y = 12, ps = 5, JO4 = 11, 
 and a; = 6, y = 14, ^3 = 7, p^ = 13. 
 
 That simple groups actually exist corresponding to the 
 two orders 660 and 1092 thus arrived at, is shewn in § 221. 
 There is also in each case a single type ; the verification 
 of this statement is left to the reader. 
 
 259. As has already been stated at the beginning of the 
 present chapter, the solution of the general problem of pure 
 group-theory, namely, the determination of all possible types of 
 group of any given order, depends essentially on the previous 
 determination of all possible simple groups. A complete solution 
 of this latter problem is not to be expected ; but for orders, which 
 do not exceed some given limit, the problem may be attacked 
 directly. The first determination of this kind was due to 
 Herr O. Holder', who examined all possible orders up to 200 : 
 he proved that none of them, except 60 and 168, correspond to 
 a simple group. Mr Cole'' continued the investigation, by 
 examining all orders from 201 to 660, with the result of shewing 
 that in this interval the only orders, which have simple groups 
 corresponding to them, are 360, 504 and 660. The existence 
 of a simple group of order 504 had not been recognised 
 before Mr Cole's investigation. 
 
 The author'' has carried on the examination from 661 to 
 1092, with the result of shewing that 1092 is the only number 
 in this interval which is the order of a simple group. 
 
 As the limit of the order is increased, such investigations 
 as these rapidly become more laborious, since a continually in- 
 creasing number of special cases have to be dealt with. There 
 is little doubt however but that, with the aid of the theorems 
 proved in the present chapter, the investigation might be 
 continued without substantial difficulty up to 2000. 
 
 1 Math. Ann. Vol. xl, (1892), pp. 53-88. 
 
 ^ American Journal of Mathematics, Vol. xv, (1893), pp. 303-315. 
 
 ■* Proo. London Math. 80c. , Vol. xxvi, (1895), pp. 333-338. 
 
260] DO NOT EXCEED 660 371 
 
 We shall here be content with verifying the result of Herr 
 Holder's and Mr Cole's investigations for orders up to 660. The 
 method used in dealing with particular numbers may suggest 
 to the reader how the determination might be continued. 
 
 260. It follows from Theorem XII that, if an odd number 
 can be the order of a simple group, it must be the product of at 
 least 6 prime factors. Moreover, by previous theorems, we have 
 seen that 3«, S^p and Sy cannot be orders of simple groups. 
 Hence certainly no odd number less than 3'' . 5 . 7 or 2835^ can be 
 the order of a simple group. Therefore, by Theorem XI, we need 
 only examine numbers less than 660 which are divisible by 
 12, 16, or 56. When each number less than 660 which is 
 divisible by 12, 16, or 56 is written in the form pi'^Pi^..., it 
 will be found that, with eleven exceptions. Theorems II to 
 XII of the present chapter immediately shew that there are no 
 simple groups corresponding to them. The exceptions are 60, 
 168, 240, 336, 360, 4.80, 504, 528, 540, 560, 660. We will first 
 deal with such of these numbers as do not actually correspond 
 to simple groups. 
 
 240 = 2^ 3 . 5. 
 A simple group of this order would contain 6 or 16 sub-groups 
 of order 5. If it contained 6 sub-groups of order 5, the group 
 could be expressed as a transitive substitution group of degree 
 6 ; and there is no gi-oup of degree 6 and order 240. If there 
 were 16 sub-groups of order 5, each would be self-conjugate in 
 a group of order 15, which must be cyclical. The group then 
 would be a doubly transitive group of degree 16 and order 
 16 . 15. Such a group (§ 105) contains a self-conjugate sub- 
 group of order 16. 
 
 336 = 2^ . 3 . 7. 
 
 A simple group of this' order would contain 8 sub-groups of 
 order 7, each self-conjugate in a group of order 42. We have 
 seen (§ 146) that there is no simple group of degree 8 and order 
 8.7.6. 
 
 It is not, of course, here suggested that 2835 is the order of a simple 
 group. It may, in fact, be shewn with little difficulty that no number of the 
 form S*pq, S^g, or S^g, where p and q are odd primes, can be the order of a 
 simple group. A much higher limit than 2835 may therefore be given for the 
 order, if odd, of a simple group. 
 
 24—2 
 
3'72 SIMPLE GROUPS WHOSE ORDERS [260 
 
 480 = 2^ 3 . 5. 
 
 Since 2^ is not a factor of 5 ! , a group of order 480 must, if 
 simple, contain 15 sub-groups of order 2^ Now 1^ is not con- 
 gruent to unity, mod. 4, and therefore (§ 78) some two sub-groups 
 of order 2^ must have a common sub-group of order 2^ Such 
 a sub-group, of order 2^ must (§ 80) either be self-conjugate, or 
 it must be contained self-conjugately in a sub-group of order 
 2^3 or 2^.5. In either case the group is composite. 
 
 528 = 2^ 3.11. 
 
 There must be 12 sub-groups of order 11, each being 
 self-conjugate in a group of order 44. A group of order 44 
 necessarily contains operations of order 22, and such an opera- 
 tion cannot be represented as a substitution of 12 symbols. 
 The group is therefore not simple. 
 
 . 540 = 2= . 3' . 5. 
 
 There must be 10 sub-groups of order 3^ each self-conjugate 
 in a group of order 2 . 3^. There must also be 36 sub-groups of 
 order 5, each self-conjugate in a group of order 15. Now when 
 the group is expressed as transitive in 10 symbols, the substitu- 
 tions of order 5 consist of 2 cycles of 5 symbols each, and 
 no such substitution can be permu table with a substitution of 
 order 3. Hence the group is not simple. 
 
 560 = 2* . 5 . 7. 
 
 There must be 8 sub-groups of order 7. A transitive group 
 of degree 8 and order 8 . 7 . 10 does not exist (§ 146). 
 
 That there are actually simple groups of orders 60, 168, 360, 
 and 660, has already been seen ; the verification that there is 
 only a single type of simple group of order 360 may be left 
 to the reader. It remains then to consider the order 504. 
 
 504 = 2^3\7. 
 
 A simple group of this order must contain 8 or 36 sub-groups 
 of order 7. There is no group of degree 8 and order 504 (§ 146). 
 Hence there must be 36 sub-groups of order 7. 
 
260] DO NOT EXCEED 660 373 
 
 Again, there must be 7 or 28 sub-groups of order 3^ ; and 
 since there is no group of degree 7 and order 504 {I.e.), there 
 are 28 sub-groups of order 3^. If two of these have a common 
 operation P of order 3, it must (§ 80) be self-conjugate in a group 
 containing more than one sub-group of order 31 If the order 
 of this sub-group is 2^3^ it must contain a sub-group of order 
 2^ self-conjugately ; and therefore this sub-group of order 2^ must 
 be self-conjugate ia a group of order 2'. 3^ at least. Such a 
 gi-oup would be one of 7 conjugate sub-groups, and this is 
 impossible. Similarly, P cannot be self-conjugate in a group 
 whose order is greater than 2^ . 3^ Hence no two sub-groups 
 of order 3^ have a common operation, except identity. 
 
 There are therefore in the group 216 operations of order 7, 
 and 224 operations whose orders are powers of 3, leaving just 
 64 other operations. 
 
 Suppose now that A is an operation of order 2, self-con- 
 jugate in a sub-group of order 2^ If A is self-conjugate in a 
 group of order 2^x {x being a factor of 3^ . 7), this group must 
 contain at least x operations of odd order and therefore at least x 
 operations of the form AS, where 8 is an operation of odd order 
 permutable with A. There are also x operations of this form, 
 
 63 
 corresponding to each of the — operations conjugate to A ; 
 
 so that the whole group contains 63 operations of the form AS, 
 where S is of odd order and permutable with A, while A is one 
 of a set of conjugate operations of order 2, each of which is self- 
 conjugate in a sub-group of order 2^ Hence taking the iden- 
 tical operation with these 63 operations of even order, all the 
 operations of the group are accounted for; and there are there- 
 fore no operations whose orders are powers of 2 except those of 
 the conjugate set to which A belongs. The sub-groups of order 
 2^ are therefore Abelian groups whose operations are all of order 
 2 ; and since the 7 operations of order 2, in a group of order 2^ 
 are all conjugate in the group of order 504, they must (§ 81) be 
 conjugate in the sub-group within which the group of order 2^ 
 is self-conjugate. Hence, finally, there must be 9 sub-groups of 
 order 2», each self-conjugate in a group of order 2^ 7, in which 
 the 7 operations of order 2 form a single conjugate set. 
 
374 SIMPLE GROUPS WHOSE ORDERS [260 
 
 A simple group of order 504, if it exists, must therefore be 
 expressible as a triply transitive group of degree 9, in which 
 the sub-group of order 56 that keeps one symbol fixed is of 
 known type. From this, several inferences can be drawn. 
 Firstly, all the 63 operations of order 2 of the 9 sub-groups 
 of order "2? must be distinct, since each operation of order 2 
 keeps just one symbol fixed. There is therefore a conjugate 
 set of 63 operations of order 2, and there are no other operations 
 of even order in the group. Secondly, the 224 operations whose 
 orders are powers of 8 cannot all be operations of order 3. For, 
 if they were, there would be 8 operations of order 3 containing 
 any three given symbols in a cycle ; and therefore the group 
 would contain, operations other than identity which keep 3 
 symbols fixed. This is not the fact; the sub-groups of order 
 3^ are therefore cyclical. 
 
 Conversely, we may now shew that such a triply transitive 
 group of degree 9, if it exists, is certainly simple. The distri- 
 bution of its operations in conjugate sets has, in fact, been 
 determined as follows. There are : — 
 
 3 conjugate sets of operations of order 1 each containing 72 operations 
 
 3 „ „ 9 „ 56 „ 
 
 1 „ ,, 3 ,, Ob „ 
 
 1 „ „ 2 ,, 63 „ 
 
 A self-conjugate sub-group, that contains a single operation 
 of order 7, must contain all of them ; and such a sub-group, 
 that contains a single operation of order 9, must contain all 
 the operations of orders 3 and 9. Hence if a self-conjugate sub- 
 group contains operations of order 9, its order must be of the form 
 
 1 -t- 224 -f- 216* -I- QZy ; 
 
 and if it contains no operations of order 9, its order must be of 
 the form 
 
 1+ 56a;' -I- 216?/' + 63/; 
 
 each of the symbols on, y, of, y'j s! being either zero or unity. 
 The order must at the same time be a factor of 504. A very 
 brief consideration will shew that all these conditions cannot be 
 satisfied, and therefore that no self-conjugate sub-group exists. 
 
 That there is a triply transitive group of degree 9 and order 
 9.8.7, has already been seen in § 113 ; and the actual formation 
 
261] DO NOT EXCEED 660 375 
 
 of the substitutions generating this group will verify that it 
 satisfies the conditions just obtained. 
 
 We may however very simply construct the group by the 
 method of § 109 ; and this process has the advantage of shewing 
 at the same time that there is only one type. 
 
 There is no difficulty in constructing the sub-group of order 
 56 as a doubly transitive group of 8 symbols. If we denote by 
 s and t the two substitutions 
 
 (1254673) and (12) (34) (56) (78), 
 
 {s, t] is, in fact, a group of the desired type. If now A is any 
 
 substitution of order 2 in the symbols 1, 2, , 8, 9 which 
 
 contains the transposition (89), and satisfies the conditions 
 
 AsA = 8„ AtA = S^ASs, 
 
 where Si, 82, 83 belong to [s, t}: then (§ 109) it follows that 
 {A, s, t] is a triply transitive group of degree 9 and order 504. 
 
 Now every operation of order 2 in the group, which inter- 
 changes 8 and 9, must transform s into its inverse. Also from 
 the 9 symbols only 7 operations of order 2, satisfying these 
 conditions, can be formed ; and if one of them belongs to the 
 group, they must all belong to it. Hence there cannot be more 
 than one type of group satisfying the conditions. 
 
 Finally, it A = (89) (23) (46) (57), 
 
 then AsA= s-\ 
 
 and AtA can be expressed in the form 
 
 (1635842)^(1685437), 
 
 where the two operations (1635842) and (1685437) belong 
 to {s, t). It follows, then, that {A, s, t] is a triply transitive 
 group of degree 9 and order 504, which satisfies all the con- 
 ditions, and that there is only a. single type of such group. 
 
 261. The order of a non-soluble group must be divisible by 
 the order of some non-cyclical simple group. Moreover, if N is 
 the order of a non-cyclical simple group and if n is any other 
 number, there is always at least one non-soluble group of order 
 Nn ; for, G being a simple group of order N and H any group 
 of order n, the direct product of and if is a non-soluble group 
 
376 
 
 NON-SOLUBLE 
 
 [261 
 
 of order Nn. Herr Holder' has determined all distinct types of 
 non-soluble groups whose orders are less than 480. He shews 
 that, besides the three simple groups of orders 60, 168, and 
 360, there are 22 non-soluble groups whose orders are given by 
 the following table. 
 
 120 
 
 180 
 
 240 
 
 300 
 
 336 
 
 360 
 
 420 
 
 3 
 
 1 
 
 8' 1 
 
 3 
 
 5 
 
 1 
 
 For the proof of all these results except the first, and for 
 the very interesting and suggestive methods that lead to them, 
 the reader is referred to Herr Holder's memoir. The case 
 of a non-soluble group of order 120 is susceptible of simple 
 treatment; and we will consider it here as exemplifying how 
 composite groups may be constructed when their factor-groups 
 and the isomorphisms of the latter are completely known. 
 
 262. The composition-factors of a non-soluble group G of order 
 120 are 2 and 60. If these may be taken in either order, then G 
 contains a self-conjugate sub-group of order 2 and a self-conjugate 
 sub-group of order 60. The latter, being simple, must be an 
 icosahedral group and cannot contain the former. Hence, in this 
 case, G must (§ 34) be the direct product of an icosahedral group 
 and a grdup of order 2. 
 
 Next, suppose that the composition-factors can only be taken in 
 the order 2, 60. Then G contains a self-conjugate sub-group H of 
 icosahedral type and no self -conjugate sub-group of order 2. Hence 
 if S is any operation of G which is not contained in H, the 
 isomorphism of H which arises on transforming its operations by S 
 must (§ 165) be contragredient. It follows that G is simply isomor- 
 phic with a group of isomorphisms of H. Now H is isomorphic 
 with the alternating group of degree five, and its group of isomor- 
 phisms is therefore (g 173) simply isomorphic with the symmetric 
 group of degree five. Hence, in this case, G is simply isomorphic 
 with the symmetric group of degree five. 
 
 Lastly, suppose that the composition-factors can only be taken 
 in the order 60, 2. Then G has a self -conjugate sub-group of order 2 
 and no self-conjugate sub-group of order 60. Hence (§ 35) G has 
 
 1 "Bildung zusammengesetzter Gruppen," Math. Ann. Vol. xlvi., 
 pp. 321-422; in particular, p. 420. 
 
 (1895), 
 
263] COMPOSITE GROUPS 377 
 
 no sub-group of order 60. Suppose that A is the self -conjugate 
 operation of order 2, and arrange the operations of G in the sets 
 
 ^ I, A; S„S,A; S„S,A; ; S,„ S,,A. 
 
 Since ^ is simply isomorphic with an icosahedral group, it must 
 (§ 203) be possible to choose three sets 
 
 S',S'A; 8",S"A; S"',S"'A; 
 such that either operation of the first, multiplied by either operation 
 of the second, belongs to the third set ; while at the same time the 
 cube of either operation of the first set, the fifth power of either 
 operation of the second set, and the square of either operation of the 
 third set, belong to the set 1, A. Now if 
 
 S'' = l, 
 
 then {S'AY = A. 
 
 Hence we may assume, without loss of generality, that 
 
 S'" = 1, S'" = 1, S'S" = S'" or S"'A. 
 
 If S'" is of order 2, so also is S"'A, and G would then contain an 
 icosahedral sub-group. Hence 
 
 S"'^=\, 
 
 and S"'^ = A. 
 
 Now ^,^=1, ^/ = 1, {S,S,f=\, 
 
 is a complete set of defining relations for the icosahedral group ; so 
 that, from the symbols S-^ and S^, exactly 60 distinct products can be 
 formed. Hence from S', S", and A, where 
 
 ^'3=1, ^"^=1, {S'Sy = A, A^=l, 
 
 A being permutable both with S' and S", exactly 120 distinct 
 products can be formed. It follows that the relations just given are 
 the defining relations of a non-soluble group of order 120, which has 
 a self -conjugate sub-group of order 2 and no sub-group of order 60 ; 
 and that there is only one type of such group. We have already 
 seen (§221) that there must be at least one such type of group ; viz. 
 the group defined by 
 
 ajg' = Y'lh + ^''^21 (mod. 5). 
 
 Ex. If p is an odd prime, shew that a group, whose composition- 
 factors are 60 and p, must be the direct product of an icosahedral 
 group and a group of order/). (Holder.) 
 
 263. Let C be a composite group; and suppose that the 
 factor-groups of the composition-series of G are the two non- 
 cyclical simple groups H' and H in the order given. Then 
 
378 NON-SOLUBLE GROUPS [263 
 
 (§ 165) if iTi is the group formed of all the operations of G 
 which are permutable with every operation of H, the direct 
 product {H, Hi] of H and Hi is a self-conjugate sub-group of G. 
 Now the composition-factors of {H, Hi} are composition-factors 
 of G. Hence H must either be isomorphic with H', or it must 
 reduce to the identical operation. The latter altern'ative can 
 
 only occur if jj has a sub-group simply isomorphic with H', 
 
 L being the group of isomorphisms of H. In the cases of the 
 simple groups whose groups of isomorphisms have been in- 
 vestigated in Chapter XI, i.e. the alternating groups, the doubly 
 transitive groups of degree p" + 1 and order \p^ {p™ — 1), and the 
 triply transitive groups of degree 2" -1-1 and order 2" (2^ — 1), 
 
 jf was found to be an Abelian group. (These groups include 
 
 all the simple groups whose orders do not exceed 660.) If H 
 and H' belong to these classes of groups, then G must be the 
 direct product of simple groups simply isomorphic with H and 
 H'. For instance, a group of order 8600, whose composition- 
 factors are 60 and 60, must be the direct product of two icosa- 
 hedral groups. 
 
 This result may clearly be extended to the case of a group, 
 the factor-groups of whose composition-series are a number of 
 non-cyclical simple groups Hi, H^, ,-9m, such that no two 
 
 of them are of the same type, while for each of them jf 
 
 is soluble. Such a group must be the direct product of n 
 simple groups which are isomorphic with Hi, H^, ,Hn. 
 
 If, however, several of these groups are of the same type, the 
 inference is no longer necessarily true. Thus if G is the direct 
 product of five icosahedral groups, and if L is the group of 
 
 isomorphisms of G, the order of ^ is 2\ 5 !, and this group 
 
 contains icosahedral sub-groups. A group of order (60)^ the 
 factor-groups of whose composition-series are all of order 60, 
 is not therefore necessarily the direct product of 6 icosahedral 
 groups. 
 
NOTES TO §§ 257, 258, 260 379 
 
 Note to § 257. 
 
 The property stated in the footnote on p. 366 may be proved as 
 follows. Let aS*! and S^ be two operations of the group which are 
 not permutable with each other ; and suppose that 
 
 The sub-group of order p'^-\ which contains S^ self-conjugately, also 
 contains iSiS self-conjugately (§ 55) j therefore it contains S self- 
 conjugately. Similarly the sub-group, which contains S^ self-conju- 
 gately, contains S^'^-^ and therefore 5 self-conjugately. Hence 2, 
 being contained self-conjugately in two distinct sub-groups of order 
 p'^-\ is a self-conjugate operation. From this it follows at once that 
 
 -r- is Abelian. 
 n 
 
 Note to § 258. 
 
 The statement, on p. 368, that the congruences p^^ = 1 (mod. p^) 
 and Pa ^ 1 (mod. p^) are inconsistent, is subject to an exception 
 if Pi+Pi + ^ is a prime. When this is the case and when 
 Pi=Pi+Pi+^, a group of order PiP^p^ can only be simple, under 
 the conditions assumed in the text, if it contains {pi—V)Ps opera- 
 tions of order p^, Pi{p2—l)p3 operations of order p^, PiPi{pi — \) 
 operations of order p^, and no other operation except identity. The 
 relation 
 
 PiP^3 = {Pi- ^)Pi+Pi (P2-'^)P3+PiP2 (ft- 1) + 1 
 cannot however be satisfied ; the group is therefore composite. 
 
 Note to §260. 
 
 No simple group of odd order is at present known to exist. An 
 investigation as to the existence or non-existence of such groups 
 would undoubtedly lead, whatever the conclusion might be, to 
 results of importance ; it may be recommended to the reader as 
 well worth his attention. Also, there is no known simple group 
 whose order contains fewer than three diiferent primes. This 
 suggests that Theorems III and IV, §§ 243, 244, may be capable of 
 generalisation. Investigation in this direction is also likely to lead 
 to results of interest and importance. 
 
APPENDIX. 
 
 The technical phraseology that has been used in this book is 
 borrowed almost entirely from French or German; and far the 
 greater number of important memoirs on the subject of finite groups 
 are written in one or the other of those languages. To enable the 
 reader to refer, with as little trouble as possible, to the writings of 
 foreign mathematicians, a table is here given of the French and the 
 German equivalents of the more important technical terms. Even 
 abroad, the phraseology of the subject has not yet arrived at that 
 settled state in which every writer uses a technical term in the 
 same sense ; but the variations of usage are not very serious. In 
 his recently published Lehrhuch der Algebra, however, Herr Weber 
 has introduced or adopted several deviations from ordinary usage ; 
 the chief of these are noted, by the addition of his name after the 
 term, in the subjoined table. 
 
 Group 
 
 Abehan group 
 
 Alternating group 
 
 Complete group 
 Composite group 
 
 Factor-group 
 
 Primitive or imprimitive group 
 
 Transitive or intransitive group 
 
 Simple group 
 
 Soluble group 
 
 Substitution group 
 
 Symmetric group 
 Group of isomorphisms 
 
 Groupe 
 
 Gruppe 
 
 Groupe des operations ^changeables 
 
 Abel'sche Gruppe 
 
 Groupe altern^ 
 
 Alternierende Gruppe 
 
 Vollkommene Gruppe 
 
 Groupe compost 
 
 Zusammengesetzte Gruppe 
 
 Groupe faoteur 
 
 Factorgruppe 
 
 Groupe primitif ou non-primitif 
 
 Primitive oder imprimitive Gruppe 
 
 Groupe transitif ou intransitif 
 
 Transitive oder intransitive Gruppe 
 
 Groupe simple 
 
 Einfache Gruppe 
 
 Groupe r&oluble 
 
 Auflosbare Gruppe 
 
 Metacyclische Gruppe ' {Weber) 
 
 Groupe des substitutions or systfeme 
 
 des substitutions conjugu^es 
 Substitutionengruppe 
 Groupe symtoique 
 Symmetrische Gruppe 
 Gruppe der Isomorphismen 
 
 ' This term is used by other German writers to denote the holomorph of a 
 group of prime order. 
 
Sub-group 
 
 Conjugate sub-groups 
 Self-conjugate sub-group 
 
 FKENCH AND GERMAN TEEMS 
 Sousgroupe 
 
 381 
 
 Untergruppe 
 ~ •■ (Web 
 
 Theiler ( Weber) 
 Sousgroupes conjugufe 
 Gleichberechtigte Untergruppen 
 Conjugirte Theiler ( Weber) 
 Sousgroupe invariant 
 Ausgezeichnete Untergruppe or iu- 
 
 variante Untergruppe 
 Normaltheiler ( Weber) 
 
 Maximum self-conjugate sub-group Sousgroupe invariant maximum 
 
 Ausgezeichnete or invariante Maxi- 
 
 maluntergruppe 
 Grosster Normaltheiler ( Weber) 
 
 Characteristic sub-group Charakteristische Untergruppe 
 
 Composition-series 
 
 Composition factors 
 
 Chief series 
 Characteristic series 
 
 Suite de composition 
 Eeihe der Zusammensetzung 
 Compositionsreihe ( Weber) 
 Facteurs de composition 
 Factoren der Zusammensetzung 
 Hauptreihe 
 
 Liickenlose Eeihe oharakteristischer 
 Untergruppen 
 
 Simple isomorphism 
 
 Isomorphisme holo^drique 
 Holoedrischer or einstuflger Iso- 
 
 morphismus 
 Isomorphisme meriddrique 
 Meriedrischer or mehrstufiger Iso- 
 
 morphismus 
 Isomorphism of a group with itself Isomorphismus der Gruppe in sich 
 
 Multiple isomorphism.! 
 
 Cogredient or contragredient iso- 
 morphism 
 
 Cogredient oder contragredient Iso- 
 morphismus 
 
 Degree (of a group) 
 Order „ „ 
 
 Genus 
 
 >> )> 
 
 Degr^ 
 
 Grad 
 
 Ordre 
 
 Ordnung 
 
 Grad ( Weber) 
 
 Geschlecht 
 
 1 It is to be noticed that, if G is multiply isomorphic with H, then H is 
 ' meri^driquement isomorphe " with Gf. 
 
382 
 
 FRENCH AND GERMAN TERMS 
 
 Substitution 
 
 Circular substitution 
 Even and odd substitutions 
 Eegular substitution 
 Similar substitutions 
 Transposition 
 
 Substitution 
 
 Substitution or Buchstabenvertausch- 
 
 ung 
 Substitution circulaire 
 Cirkularsubstitution 
 Substitutions positives et negatives 
 Gerade und ungerade Substitutionen 
 Substitution r^gulifere 
 Regulare Substitution 
 Substitutions semblables 
 Ahnliche Substitutionen 
 Transposition 
 Transposition 
 
 One other term, introduced by Herr Weber, for which no simple 
 equivalent is used in this book, must be mentioned. The greatest 
 sub-group which is common to two groups he calls the "Durch- 
 schnitt " of the two groups. 
 
 The ratio of the order of a sub-group H, to the order of the group 
 G containing it, is called by French writers the " indice," by German 
 the " Index," of H in G. The phrase is most commonly used of a 
 substitution group in relation to the symmetric group of the same 
 degree. 
 
 The smallest number of symbols displaced by any substitution, 
 except identity, of a substitution group is called by French writers 
 the "classe," by Germans the "Klasse," of the group. 
 
INDEX. 
 
 {The numbers refer to pages.) 
 
 Abel, quoted, 46. 
 
 Abelian gkoup, definition of, 46 ; 
 
 sub-groups of, 47, 48, 55-69; 
 
 existence of independent generating operations of, 52 ; 
 
 invariance of the orders of a set of independent generating operations 
 of, 64; 
 
 symbol for, of given type, 55. 
 Abelian gkotip of order p" and type (1, 1, ..., 1), 58; 
 
 number of sub-groups of, whose order is given, 60 ; 
 
 number of distinct ways of choosing a set of independent generating 
 operations of, 59 ; 
 
 group of isomorphisms of, 243, 244 (see also Homogeneous uneak 
 
 GBOBP) ; 
 
 holomorph of, 245. 
 Alteenating group, definition of, 139 ; 
 
 group of isomorphisms of, 246 ; 
 
 is simple, except for degree 4, 164. 
 BooHBBT, quoted, 153. 
 BoLZA, quoted, 162. 
 
 BuKNSlDE, quoted, 65, 97, 157, 311, 317, 336, 345, 352, 366, 370. 
 Cauohy, quoted, 90. 
 Cattlet, quoted, 306. 
 Chabaoteeistic seeies, definition of, 232; 
 
 invariance of, 233 ; 
 
 of a group whose order is a power of a prime, 233-235. 
 Chaeaoieeistic sub-geoup, definition of, 232 ; 
 
 groups with no, are either simple or the direct product of simply 
 isomorphic simple groups, 232. 
 Chief composition seeies, or Chief seeies, definition of, 123; 
 
 invariance of, 123. 
 Cole, quoted, 196, 370. 
 Cole and Gloveb, quoted, 137. 
 COLOUE-GEOUPS, 306-310 ; 
 
 examples of, 310. 
 Complete geoup, definition of, 236 ; 
 
 group of isomorphisms of a simple group of composite order is a, 238 ; 
 
 holomorph of an Abelian group of order p" and type (1, 1 1) is a, 
 
 239; 
 holomorph of a cyclical group of odd order is a, 241 ; 
 symmetric group is a, except for degree 6, 246 ; 
 groups which contain a, self-oonjugately are direct products, 236. 
 
384 INDEX. 
 
 Composite groups, definition of, 29 ; 
 
 non-soluble, 376-378 ; 
 
 of even order, 360-365. 
 Composition factors, definition of, 118. 
 CoMPOsiTiON;SERiES, definition of, 118 ; 
 
 invariance of, 122 ; 
 
 examples of, 128, 129. 
 CoNJUOATE OPERATIONS, definition of, 27 ; 
 
 complete set of, 31. 
 CoNJDGATE SUB-GROUPS, definition of, 29 ; 
 
 complete set of, 32 ; 
 
 operations common to or permutable with a complete set of, form a 
 self-conjugate sub-group, 33. 
 Dedekind, quoted, 89. 
 Defining relations of a group, definition of, 21 ; 
 
 limitation on the number of, when the genus is given, 283 ; 
 
 for groups of genus zero, 291 ; 
 
 for groups of genus unity, 301, 302 ; 
 
 for groups of orders p', p^, p^, 87, 88, 89 ; 
 
 for groups of order pq, 100 ; 
 
 for groups of order p^q, 133-137 ; 
 
 for groups whose orders contain no square factor, 354 ; 
 
 for the holomorph of a cyclical group, 240, 241 ; 
 
 for the symmetric group of degree 5, 305; 
 
 for the simple group of order 168, 305. 
 Degree of a substitution group, definition of, 138; 
 
 is a factor of the order, if the group is transitive, 140. 
 Direct product of two groups, definition of, 40 ; 
 
 represented as a transitive group, 190. 
 Direct product of two simply isomorphic groups of order n represented as 
 
 a transitive group of degree n, 146, 147. 
 Doubly transitive groups, generally contain simple self-conjugate sub-groups, 
 192; 
 
 the sub-groups of, whio]! keep two symbols fixed, 212 ; 
 
 with a complete set of triplets, 214 ; 
 
 of degree n and order n(n-l), 155-157. 
 Dyck, quoted, 22, 44, 172, 195, 255, 292, 301, 305. 
 Dyck's theorem that a group of order n can be represented as a regular 
 
 substitution group of degree n, 22-24. 
 Factor groups, definition of, 38 ; 
 
 set of, for a given group, 118 ; 
 
 invariance of, 118. 
 Forsyth, quoted, 280, 283. 
 Fractional linear group, definition of, 311 ; 
 
 analysis of, 319-333 ; 
 
 generalization of, 334, 335. 
 Frobenius, quoted, 39, 45, 65, 71, 90, 97, 108, 110, 114, 166, 232, 250, 251, 345, 
 
 354, 358, 360. 
 Frobenius and Stickeleerger, quoted, 46. 
 
 Frobenius's theorem that, if n is a factor of the order of a group, the 
 number of operations of the group whose orders divide n is a 
 multiple of n, 110-112. 
 Galois, quoted, 192, 334. 
 
 General discontinuous group with a finite number of generating operations, 
 256; 
 
 relation of special groups to, 257, 258. 
 Genus of a group, definition of, 280. 
 Giersteb, quoted, 311. 
 Graphical representation, of a cyclical group, 260, 261, 288; 
 
 of a general group, 262-266 ; 
 
 of a special group, 266-269; 
 
INDEX. 385 
 
 Graphical bepeeseniation, of a group of finite order, 273-278 ; 
 
 examples of, 269-272, 279, 288-290, 294, 296, 299, 300, 303, 310. 
 Group, definition of, 11, 12 ; 
 
 continuous, discontinuous, or mixed, 13, 14 ; 
 
 order of, 14; 
 
 multiplication table of, 20, 49; 
 
 fundamental or generating operations of, 21 ; 
 
 defining relations of, 21 ; 
 
 cyclical, 25 ; 
 
 simple and composite, 29 ; 
 
 Abelian, 46 ; 
 
 soluble, 130 ; 
 
 symmetric, 139; 
 
 alternating, 139 ; 
 
 complete, 236 ; 
 
 group of isomorphisms of a, 223 ; 
 
 general, 256; 
 
 special, 257 ; 
 
 quadratic, 326 ; 
 
 dihedral, 287 ; 
 
 tetrahedral, 289 ; 
 
 octohedral, 289 ; 
 
 icosahedral, 289 ; 
 
 holomorph of a, 228 ; 
 
 genus of a, 280 ; 
 
 symbol for a, 27 ; 
 
 (see also Substitution group). 
 Group of isomorphisms, contains the group of cogredient isomorphisms 
 self-conjugately, 226; 
 
 of a cyclical group, 239-242; 
 
 of an Abelian group of order p" and type (1, 1, , 1), 243, 244, 311- 
 
 317, 336-338; 
 
 ofXthe alternating group, 246; 
 
 of doubly transitive groups of degree p^' + l and order ip"{p^-l), 246- 
 248; 
 Groups of genus zero, 286-292; 
 
 of genus one, 295-302; 
 
 of genus two, 302. 
 Groups whose order is p", where p is a prime, 61 et seq. ; 
 
 always contain self -conjugate operations, 62; 
 
 every sub-group of, is contained self-oonjugately in a greater sub-group, 
 66; 
 
 number of sub-groups of given order is congruent to unity, mod. p, 71 ; 
 
 case in which there is only one sub-group of a given order 72-75; 
 
 number of types which contain self-conjugate cyclical sub-groups of 
 order p'-^, 76-81 ; 
 
 determination of distinct types of orders p'^, p^, p*, where p is an odd 
 prime, 81-88 ; 
 
 table of distinct types of orders 8 and 16, 88, 89. 
 Groups whose orders contain, no squared factor, 353, 354; 
 
 no cubed factor, 359. 
 Groups whose sub-groups of order p" are all cyclical, 352, 353. 
 Hepeter, quoted, 216. 
 
 Holder, quoted, 38, 87, 119, 137, 224, 236, 246, 249, 344, 353, 370, 376. 
 Holomorph, definition of, 228 ; 
 
 of a cyclical group, 240-242; 
 
 of an Abelian group of order p" and type (1, 1, , 1), 245. 
 
 Homogeneous linear group, definition of, 244; 
 
 composition series of, 314-317; 
 
 represented as a transitive substitution group, 336, 337. 
 
 B. 25 
 
386 INDEX. 
 
 HoMoaBNEous LINEAR GKoup, simple groups defined by, 338, 339; 
 
 generalization of, 340-342. 
 HuEwiTz, quoted, 24, 280, 282. 
 Idbntioal opeeation, definition of, 12. 
 Impbimitive oeoups, definition of, 171 ; 
 
 of degree 6, 181, 182. 
 Impeimitive systems, definition of, 172; 
 
 of a regular group, 173; 
 
 of any transitive group, 176 ; 
 
 properties of, 185, 186. 
 Impeimitive self-conjuoate sub-oroup of a doubly transitive group, 193, 194. 
 Inikansitiye groups, definition of, 140 ; 
 
 properties of, 159-162; 
 
 transitive sets of symbols in, 159, 166; 
 
 of degree 7 with transitive sets of 3 and 4, 163, 164. 
 Isomorphism of two groups, simple, definition of, 32 ; 2 ?- 
 
 multiple, definition of, 36 ; ' 
 
 general, definition of, 41. 
 Isomorphisms of a group with itself, definition of, 222 ; 
 
 cogredient and oontragredient, definition of, 224 ; 
 
 class of, definition of, 224 ; 
 
 limitation on the order of, 252. 
 Isomorphism of general and special groups, 257, 258. 
 Jordan, quoted, 46, 119, 143, 146, 149, 150, 153, 189, 196, 198, 204, 311. 
 Klein, quoted, 224, 289, 292. 
 Klein and Feicke, quoted, 162. 
 Limitation on the order of a group of given genus, 282; 
 
 on the number of defining relations of a group of given genus, 283. 
 Maillet, quoted, 108. 
 Marggraee, quoted, 198. 
 Masohke, quoted, 310. 
 Mathieu, quoted, 155. 
 Maximum sub-group, definition of, 35. 
 Maximum self-conjugate sub-group, definition of, 35. 
 MiLLEE, quoted, 105, 196, 259. 
 Minimum self-oonjugate sue-group, definition of, 124; 
 
 is a simple group or the direct product of simply isomorphic simple 
 groups, 127. 
 MooRE, quoted, 215, 336. 
 Multiplication table of a group, 20, 49. 
 Multiply isomorphic groups, definition of, 36. 
 Netto, quoted, 141, 198, 216. 
 
 Number of symbols unchanged by aU the substitutions of a group, 165, 166. 
 Operations common to two groups form a group, 27 ; 
 
 of a group, which are permutable with a given operation or sub-group, 
 form a group, 29, 31 ; 
 
 common to or permutable with each of a complete set of conjugate sub- 
 groups form a self-conjugate sub-group, 33. 
 Oedee of a group, definition of, 14; 
 
 of an operation, definition of, 14. 
 Permutable operations, definition of, 12; 
 Permutable groups, definition of, 41. 
 
 Permutabilitt of an operation with a group, definition of, 29. 
 Primitive groups, definition of, 171 ; 
 
 when soluble, have a power of a prime for degree, 192 ; 
 
 limit to the order of, for a given degree, 199 ; 
 
 with a transitive sub-group of smaller degree, 197, 198 ; 
 
 of prime degree, 201 ; 
 
 of degrees 3, 4 and 5, 205 ; 
 
 of degree 6, 205, 206 ; 
 
INDEX. 387 
 
 Primitive oeotjps, of degree 7, 206-208; 
 
 of degree 8, 209-211. 
 Primitivitt, test of, 184. 
 
 Eegulae division of a surface, representation of a group by means of, 278. 
 Eepeesentation of a gboup, in transitive form ; i.e. as a transitive substitu- 
 tion group, 22, 173-179 ; 
 
 in primitive form, 177. 
 Eepkesentation, graphical (see Geaphioal eepeesentation). 
 Self-conjugate opeeation, definition of, 28 ; 
 
 of a group whose order is the power of a prime, 62 ; 
 
 of a transitive substitution group must be a regular substitution, 144. 
 Self-conjugate sub-group, definition of, 29 ; 
 
 generated by a complete set of conjugate operations, 34; 
 
 of a primitive group must be transitive, 187 ; 
 
 of an imprimitive group, 187, 188; 
 
 of a ft-ply transitive group is, in general, (/b-l)-ply transitive, 189. 
 Simple geoups, definition of, 29; 
 
 whose orders are the products of not more than 5 primes, 367-370 ; 
 
 whose orders do not exceed 660, 370-375. 
 Simply isoisioRPHio groups, definition of, 22 ; 
 
 said to be of the same type, 22. 
 Soluble groups, definition of, 130 ; 
 
 properties of, 131, 132 ; 
 
 special classes of, 345-359. 
 SuB-OEOUP, definition of, 25 ; 
 
 order of a, divides order of group, 26 ; 
 
 (see also, conjugate, self-conjugate and ohabacteeistio sub-geoup). 
 Substitution, definition of, 1; 
 
 cycles of, 2; 
 
 identical, 4; 
 
 inverse, 4; 
 
 order of, 6; 
 
 circular, 7; 
 
 regular, 7 ; 
 
 similar, 7; 
 
 permutable, 8; 
 
 even and odd, 10; 
 
 symbol for the product of two or more, 4. 
 Substitutions which are permutable, with a given substitution, 215, 216; 
 
 with every substitution of a given group, 145, 146, 217 ; 
 
 with every substitution of a group, whose degree is equal to its order, 
 form a simply isomorphic group, 146. 
 Subsiitution group, degree of, definition of, 138; 
 
 regular, definition of, 24; 
 
 transitive and intransitive, definition of, 140; 
 
 primitive and imprimitive, definition of, 171; 
 
 multiply transitive, definition of, 148; 
 
 representation of any group as a regular, 22; 
 
 degree of transitive, is a factor of its order, 140; 
 
 order of a ft-ply transitive, whose degree is n, is a multiple of 
 n(n-l) {n-li + l), 148; 
 
 construction of multiply transitive, 150; 
 
 limit to the degree of transitivity of, 152; 
 
 transitive, whose substitutions, except identity, displace all or all but 
 one of the symbols, 141-144; 
 
 doubly transitive, of degree n and order m(n-l), 155-157; 
 
 triply transitive, of degree n and order ra(re-l)(ji-2), 158; 
 
 quintuply transitive, of degree 12, 170; 
 
 conjugate substitutions of, are similar, 144; 
 Substitution geoups whose orders are powers of primes, 218, 219. 
 
388 INDEX. 
 
 Sylow, quoted, 62, 90. 
 SitLOw's THEOEEM, 92-95; 
 
 some direct consequences of, 97-100 ; 
 extension of, 110. 
 Symbol for the product of two or more substitutions, 4; 
 for the product of two or more operations, 12; 
 for a group generated by given operations, 27; 
 &(P) and e(P), definition of, 250, 251. 
 Symmeikio gkoup, definition of, 139; 
 
 of degree n has a single set of conjugate sub-groups of order (n-l)l 
 
 except when ra is 6, 200; 
 of degree 6 has 12 simply isomorphic sub-groups of order 51 which 
 
 form two distinct conjugate sets, 200; 
 is a complete group, except for degree 6, 246. 
 Teansfokming an operation, definition of, 27. 
 Transitive geoup, definition of, 140; 
 
 representation of any group as a, 175-179; 
 
 number of distinct modes of representing the alternating group of 
 degree 5 as a, 179, 180. 
 Transpositions, definition of, 9; 
 
 representation of a substitution by means of, 9; 
 
 number of, which enter in the representation of a substitution is either 
 always even or always odd, 10. 
 Triply transitive groups of degree to and order re (n - 1) (re - 2) , 158. 
 Type of a geoup (see Simply isomorphic geoups). 
 
 Types or group, distinct, whose order is p", p^, or p^, where p is an odd 
 prime, 87, 88; 
 8 or 16, 88, 89; 
 
 pq, where p and q are different primes, 100, 101 ; 
 p^q, where p and q are different primes, 133-137 ; 
 24, 101-104; 
 60, 105-108. 
 Wbbbb, quoted, 46. 
 Young, quoted, 63, 68, 87, 310. 
 
 CAMBRIDGE; PRINTED BY J. AND 0. F. CLAY, AT THE UNIVERSITY PEESS.