CORNELL UNIVERSITY LIBRARIES Mathematica Library White Hall 3 1924 058 531 801 DATE DUE GAYLORD PRINTEDINU.S.A. Ui ^y. Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924058531801 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1990. ^•7 SEP i'- c. / / / publications 'f u?;' Ol' THE lllniversit^ of Pennsylvania SERIES IN MATHEMATICS No. 3 GROUPS OF ORDEll p"" WHICH CONTAIN CYCLIC SUBGROUPS OF ORlj^ER p" tOi-s Lp:\viy un'LXG neikirk SOMKTl.ME lIAKTilSON HKSK.AllCir ITCI.I.OW IN MATH KM ATIl- Fiiblixhed for the Cnirrr^itij I'HILAWai'iriA THE JOHN C. WINSTON CO., Ar.Exxs ](H)ii AiiCii Stkef.t, Pnu.AiiKi.i'm.i, Pa, 1900 '^■b\ IPublications TUniversit^ of pennsiglvania SERIES IN MATHEMATICS No. 3 GROUPS OF ORDER p"' WHICH CONTAIN CYCLIC SUBGROUPS OF ORDER p— ^ BY LEWIS IRVING NEIKIRK SOMETIME HABBISOK BESEABCH FELLOW IN MATHEMATICS Putliahed for the University PHILADELPHIA THE JOHN C. WINSTON" CO., Agents 1006 Aech Street, Philabelphia, Pa. 1905 LIbKAKI MAY 1 7 1939 INTRODUCTORY NOTE. This monograph was begun in 1902-3. Class I, Class II, Part I, and the self-conjugate groups of Class III, which contain all the groups with independent generators, formed the thesis which I presented to the Faculty of Philosophy of the University of Pennsylvania in June, 1903, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The entire paper was rewritten and the other groups added while the author was Research Fellow in Mathematics at the University. I wish to express here my appreciation of the opportunity for scientific research afforded by the Fellowships on the George Leib Harrison Founda- tion at the University of Pennsylvania. I also wish to express my gratitude to Professor George H. Hallett for his kind assistance and advice in the preparation of this paper, and espe- cially to express my indebtedness to Professor Edwin S. Crawley for his support and encouragement, without which this paper would have been impossible. Lewis I. Neikirk. University or Pennsylvania, Mny, 1905. GROUPS OF ORDER jp", WHICH CONTAIN CYCLIC SUBGROUPS OF ORDER p^-^* IIY LEWIS IRVING NEIKIRK Introduction. The groups of order p^, which contain self-conjugate cyclic subgroups of orders ^"~' , and p""* respectively, have been determined by BuBN- slDE,t and the number of groups of order p", which contain cyclic non- self-conjugate subgroups of order jd"'~' has been given by Miller. \ Although in the present state of the theory, the actual tabulation of all groups of order p^ is impracticable, it is of importance to carry the tabu- lation as far as may be possible. In this paper all groups of order p"" (^p being an odd prime) which contain cyclic subgroups of order p"^^ and none of higher order are determined. The method of treatment used is entirely abstract in character and, in virtue of its nature, it is possible in each case to give explicitly the generational equations of these groups. They are divided into three classes, and it will be shown that these classes correspond to the three partitions: (m— 3, 3), (m — 3, 2, 1) and {m — 3, 1, 1, 1), of m. We denote by G^ an abstract group G of order p"" containing operators of order p'^~^ and no operator of order greater than ^"'~* . Let P denote one of these operators of G of order p^~^ . The p^ power of every oper- ator in G is contained in the cyclic subgroup { -P } , otherwise G would be of order greater than p*" . The complete division into classes is efEected by the following assumptions : I. There is in G at least one operator Q^ , such that (^ is not con- tained in { P } . II. The jo' power of every operator in G is contained in { -P } , and there is at least one operator Q^, such that Q'l is not contained in * Presented to the American Mathematical Society April 25, 1903. t Theory of Groups of a Finite Order, pp. 75-81. j Transactions, vol. 2 (1901), p. 259, and vol. 3 (1902), p. 383. 5 6 NEIKIEK : GROUPS OF OBDEE p^, WHICH III. The ^th power of every operator in G is contained in { -P } • The number of groups for Class I, Class 11, and Class III, together with the total number, are given in the table below : I II. II, 11, n III Total P>3 m>8 9 20+^ 6 + 2p 6 + 2^ 32 + 5p 23 23 64 + 5;j 63 + 5p 61 + 5^) 72 f>3 m=8 8 20+;? 6 + 2^ 6 + 2^ 32 + hp P>3 m=7 6 20 +^ 6 + 2p 6 + 2^ 32 + 5p 23 p = 3 m>8 9 23 12 12 47 16 p=3 m=8 8 23 12 12 47 16 71 p=3 7n=7 6 23 12 12 47 16 69 Class I. 1. General notations and relations. — The group G is generated by the two operators P and Q^ . For brevity we set * Q',P'QlP'..-=[a,h,c,d,-..-\. Then the operators of G are given each uniquely in the form We have the relation (1) Qf=P*^'. There is in & , a subgroup ff^ of order ^"~', which contains { P } self- conjugately.f The subgroup IT^ is generated by P and some operator Q^P' of G; it then contains Q^ and is therefore generated by P and Qf; it is also self-conjugate in ^j = { §J , P } of order ^""S and ^ is self- conjugate in G . From these considerations we have the equations (2) t Qt^ P §f = P'+*'^, *With J. W. YOTJNG, On a eertain group of isomorphisms, American Jonrnal of Mathematics, vol. 25 (1903), p. 206. t BUENSIDK : Theory of Grovps, Art. 54, p. 64. tlbid., Art. 56, p. 66. CONTAIN CYCLIC STTBGEOtrPS OF ORDEE p*""'. 7 (3) er''i'«f=Cf^P"s (4) Qr'PQ^=Q^JP^^- 2. Determination of H^. Derivation of a formvla for [yp^, a;]'. — From (2), by repeated multiplication we obtain and by a continued use of this equation we have [-y/,x,y/]= [0,x{l+hp'--*y^ = [0,x(H- %>"-*)] (m>4) and from this last equation, (5) iyp\ x]'= lsyp\x{a + Tc{;)yp--']'\. 3. Determination of H^. Derivation of a formula for [^yp , x'] ' ■ — It follows from (3) and (5) that [-^, 1,/] = [ff'l^^f, ,^ {i + f ^.p-]] (»><). Hence, by (2), /8j5^p'^0(modp'), 1 From these congruences, we have for m > 6 a\ = 1 (modp'), Oj = 1 (modp'), and obtain, by setting aij = 1 + a^p\ the congruence ^^"^ a^"^ (^'^ + '^^^^^ ^ ^P^* (modj.-') ; and so (a, + A/3)p'=0(modp-*), since (^ + °<)^-^^l(modp^). From the last congruences (6) (a, + A/S)/ = kp"^ (modp"-') . 8 NEIKIRK : GROUPS OF ORDER jp", WHICH Equation (3) is now replaced by (7) Q-'F Q-p = Qf^ pi+«sp» . From (7), (5), and (6) [- yp, X, yp'\ = [^zyp\ x{l+ a^yp^ } + M{l)yp'"-*]. A continued use of this equation gives [yp,xy=lsyp+fi{l)xyp\ xs + {l){a^xyp' + 0k{l)yp^*}+^k{l)x'yp'^-*'{. 4. Determination of G. — From (4) and (8), From the above equation and (7), o^ = 1 (modp^), Oj = 1 (modj9). Set Oj = 1 + Oj^ and equation (4) becomes (9) §j-'P§, = q^vp^^<^v. From (9), (8) and (6) [-y, l.i-T = P-iJ^^^V, (1 + «.i»"], and from (1) and (2) (1 + a^pY + ^j^ (l + g2J')'' -1 ^ ^ J ^ kp'^-* (mod p'^^). By a reduction similar to that used before, (10) ( flj + JA )j9' = ifcp"*-* ( mod js'"-' ) . The groups in this class are completely defined by (9), (1) and (10). These defining relations may be presented in simpler form by a suitable choice of the second generator Q^ . From (9), (6), (8) and (10) [l,a;]^=[/,.Tp'] = [0,(a; + A)/] (m>6), CONTAIN CYCJLIC SUBGROUPS OF OEDEE p'^~^. 9 and, if X be so chosen that X + h = (mod^"^*), Qi P' is an operator of order p^ whose p^ power is not contained in {-?*}• Let Q^P' = Q. The group G is generated by Q and P, where Placing A = in (6) and (10) we find a^jo' = a^p^ = kp^* (moA. p""-^) . Let flj = ap"-' , and a^^ap'^'' - Equations (7) and (9) are now replaced by (11) As a direct result of the foregoing relations, the groups in this class correspond to the partition (to — 3 , 3 ) . From (11) we find [-y,l,y'\ = \_byp,l + ayp'^''\ (m>8).* It is important to notice that by placing y =p and p^ in the preceding equation we find that I = /9(modp), a= a= Jc {mod. p^){m> 7).t A combination of the last equation with (8) yields [-2/. a;. 2/] = [.^yp + i\l)yp\ (12) x(l + ayp'^^) + ah{l)yp'^^ + a6'(|)yp"'-'] (.n>8).t From (12) we get * For m = 8 it is neceaaaty to add a' ( a )i'' to the exponent of P and for m = 7 the terms a{a-ir abp/2 ) {') p^ + a" (_l) p' to the exponent of P, and the term ah i' ) p' to the exponent of Q. The extra term Zlab'k ( 3 ) is to be added to the exponent of P for m-=7 and J) = 3. tForTn = 7, nj^ — a'p'l2 = ap'{moip'), op' = tp' ( mod pM . Form = 7and p = Z the first of the above congraences has the extra terms 27 ( a* + "^/^^ ) on the left side. X For m =: 8 it is necessary to a' + a' ( 3 )!>' } to the exponent of P, with the extra term 27 db^k (_>) xIot p=^3, and the term ab{\)xp^ to the exx>onent of Q. 10 NEIKIEK : GROUPS OF ORDER p", WHICH (13) =^ + '^y{{x + b{l)p + ¥{l)p^){l) + ihx'p + 2b'x{l)p'){l) + bx\l)p'}p-'] (m>8).» 5. Transformation of the Groups. — The general group G of Class I is specified, in accordance with the relations (2) (11) by two integers a, b which (see (11) ) are to be taken mod p^, mod p^, respectively. Accord- ingly setting a = aj^\ b^b^p", where dv\_a^,p] = \,dvl\,p] = 1 (;i=o,l,2, 3;/«=0, 1,2), we have for the group G = G{a, b) = G{a, b){P, Q) the generational determination : G(a,b): Not all of these groups however are distinct. Suppose that G{a,b){P,Q)-^Gia',b')(F',Q'), by the correspondence where Q[ = Q'yp'-'p'^, and P[ = Q'^F'^ with y' and x prime to p . Since Q-^PQ= Qippn-P— , then Qr'P[ Q[ = Qi^'P;'^"^, *Forjn = 8iti8 necessary to add the term ia»y (s) [iy(2a — 1) — l]p* to the ex- ponent of P, and foi m = 7 the terms + -2-(0— 2-i''+-2-(^ 77 ^y-COjap* }■ 27aJay{|^[(J)y'-(2»-l)y-h2](;)+«(6"Jfc + a')(2y'-|-l)(;)) nd the terms with the extra terms for ; = 3 , to the exponent of P, and the terms ab 2 to the exponent of Q CONTAIN CYCLIC SUBGllOXJPS OF ORDEE /)"-'. 11 or in terms of Q', and P' [y + b'xy'p + V\%)y'p\ x{l + a'y'p"^-') + a'b'(^,)y'p'^-' + ah'^Dyp"-"} = [y+by'p, x + {ax+ bx'p)p'^-'] (m>8) and (14) by' = b'xy' + b'^{l)y'p (mod^=), (15) ax + bx'p= a'y'x + a'b'{l)y'p + a'b'\l)y'p'' (mod jj'). The necessary and sufficient condition for the simple isomorphism of these two groups G{a, b) and G(a', b') is, that the above congruences shall be consistent and admit of solution for cc, y , x' and y'. The congruences may be written 6,^9" = b[xp>^' + b[\l)p''>^'+' (mod/), a,xp''+b^x'p>^+^=y'{a[xp>''+a[b[{l)p>''+>''+^+a[b[\l)p>''+^'''+^}{modp'). Since dv[_x, p] =1 the first congruence gives /* = /*' and x may always be so chosen that b^ = l. We may choose y' in the second congruence so that X=\' and a^ = l except for the cases X'= fi + l = fi + 1 when we will so choose x' that X = 3. The type groups of Class I for m > 8 * are then given by (I) G{p\ jj" ) : Q-'FQ= Qp>+'^pi+p'»-«+\ Qp* = 1 , i*"-' = 1 //x = 0, 1,2; A = 0, l,2;A^^;\ \/ix = 0, 1,2; A = 3 /■ Of the above groups G {p^ , p'^ ) the groups for /i = 2 have the cyclic subgroup {P] self-conjugate, while the group G{p^,p^) is the abelian group of type ( wi — 3 , 3 ) . Class II. 1. General relations. There is in G^ an operator Q^ such that Qf is contained in {P} while Q^ is not. (1) Qf = P*''. * For m = 6 the additional term ayp appears oji the left side of the coDgraence (14) and 0{1, p*) and G{1, p) become simply isomorphic. The extra terms appearing in con- graenoe (15) do not effect the resnlt. For m=: 7 the additional term ay appears on the left side of(14)andG(l,l), 0{1, p), and G{l,p') become simply isomorphic, also G(p,p)aui6{p,p'). 12 NEIKIKK : GROUPS OF OEDEE p", WHICH The operators Qj and P either generate a subgroup H^ of order />"■"', or the entire group G . Section 1. 2. Groups with independent generators. Consider the first possibility in the above paragraph. There is in ^ , a subgroup H^ o£ order ^°'~^, which contains {P] self-conjugately.* H^ is generated by Q\ and P. H^ contains H^ self-conjugately and is itself self -conjugate in G. From these considerations (2) Q-^PQ\ = P'*>^-\^ (3) q-'pq = qi^p^^. 3. Determination of H^ and H^. From (2) we obtain (4) \_yp,xy=isyp,x{s-\-]c{\)yp'^*]'\ (m>4), and from (3) and (4) A comparison of the above equation with (2) shows that ^£i^;, = 0(mod;,= ), "'■ { ^ + ? ^^l^*""' } + f^ ^hp = l + Icp--^ (modp-'), and in turn a^ = l(mod/), aj=l(modjj) (ro>5). Placing flj = 1 -f a^p in the second congruence, we obtain as in Class I (5) {a^ + ^h)p''=kp^-'{viiodip'-^^) (m>5). Equation (3) now becomes (6) Q-'PQ^=Q»P'+''^. The generational equations of H-^ will be simplified by using an operator of order p^ in place of Q^ . * BUENSIDK, Theory vf Groups, Art. 54, p. 64. \Ihid., Art. 56, p. 66. CONTAIS CYCLIC SUBGEOtTPS OF OKDEE jj*""'. 13 From (5), (6) and (4) in wmcn + iaklls{s-l){2s-l)y^-{l)y]x}p-\ Placing s=p'^ and y = 1 in the above If X be so chosen that (23 + A) = (mod^"-") (»n>5) Q^P' will be the required Q of order p^. Placing A = in congruence (5) we find a^y = %)"'-^(mod^°'-'). Let flj = ap^-^. H^ is then generated by (7) Q-l P y = §P^ pi+ap"-' . Two of the preceding formulae now become (8) [-y, «, y] = [/Sxj(p, x{\ + ajT)™-*) + /9A;(|)y^— ■•], (9) [y, »]'= [sy + U,p, xs + T^..^"-'], where and ^. = «(0»'y + ^M(0(D + (3)=^^}yi> (m>6).* 4. Determination of G . Let 5j be an operation of G not in ^j. ^ is in ^j. Let (10) Pf=^'P''^. Denoting B'[^P'R\Q;P^ -by the symbol [a, &, c, cZ, e,/, ■ ■ • J, all the operations of G are contained in the set [ » , y, a; ] ; z=0,l,2,...,^-l;y = 0,l,2,..,iJ^-l;x = 0,l,2,-.-,p"'-'-l. * For m = 6 it is necessary to add the terms rtfc f 8(8— l)(2s — 1) { '''-^\'^-'\ ^-i.\)vU^w.. 14 NEIKIRK : GKOUPS OF ORBEK p", WHICH The subgroup J^ is self-conjugate in G . From this (11) E-'PR,= (^J^\ (12) E-^ QR^ = Q^' P"'^"-* .* In order to ascertain the forms of the constants in (11) and (12) we obtain from (12), (11), and (9) [-ij, 1, 0,^] = [0, cZ? -f Jyp, iV^)-']. By (10) and (8) R^ QRl = P-"^ QP"^ = QP-'^i^'"^. From these equations we obtain d\=l (^vaoAp) and dj = l(modj)). Let d^ = 1 + dp. Equation (12) is replaced by (13) P7I QR^ = Q'+* P"""^ . From (11), (13) and (9) [-P, 0, 1,;,] = [^lrl6, + ^, a? + ^i^"-] in which By (10) and (8) R-^PRl = q-^fp(^f = pi+oAp—, and from the last two equations a^ = 1 (modj?*"-') and a, = 1 (mod;?'"-^) (wi>6); Oj = 1 (modjj) (m = 6). and Placing Cj = 1 -I- Oj^"-* (m > 6 ) ; Cj = 1 -f a^p (m = 6 ). ^=0 (modj9),t ^^6, = 6j^ = (mody), 6, = (mod^j). 1 Let 6j = 6p and we find of = 1 (mod^"-^), flj = 1 (mod;3"^»). *BnBNSIDK, Theory of Groups, Art. 24, p. 27. t-E'has an extra term for 7n = 6 and p = 3, which rednoea to 36, Cj- This does not affect the reasoning except f or Ci = 2 . In this case change P« to P and c, heoomes 1 . CONTAIN CYCLIC BUBGEOITPS OF OKDEE j}"'"'. 15 Let Oj = 1 + Ojj)"^* and equation (11) is replaced by (14) Ii-'PB,= Qf^F'+'"P"''. The preceding relations will be simpliBed by taking for jB, an operator of order p . This will be effected by two transformations. From (14), (9) and (13) [1,2/]"= ^p, yp, ^^p-'^ = [O, {\ + y)p, MP --^P"-^],* and if J/ be so chosen that X + y = (modp), ^j = i?j ^ is an operator such that ^ is in { P } . Let ^ = P"". Using Sj in the place of JR^ , from (15), (9) and (14) p, o,xp + -^p'^-*\ = 1 0, 0, (x+ Oi' + -fi'"' J' and if a; be so chosen that ax X + l+YP"'-' = 0{modp''-'), then S = SjJP' is the required operator of order ^. J?' = 1 is permutable with both Q and P . Preceding equations now assume the final forms (15) Q-'PQ= Q^pp>+'?-', (16) B-'FJR = gvj^i+'p-^^ (17) B-' QR = q^+^rp^p^ , with 2?p = 1, Q"' = 1, P^' = 1. The following derived equations are necessary (18) [0, -y, x, 0, y] = [0, ^xyp, a;(l + ayp^-') + a/3(^)yp-'-'] ,t (19) [-2/,0,x, -y]= [0,6xyp,x(l + ayp"'-*) + a6(^)yp'»-^], (20) [-y,a;,0,y]= \_Q,x{l + dyp), cxyp'^-''\. * Ihe extra terms appearing in the exponent of P for m = 6 do not alter the tesnlt. t For m = 6 the term a^ ( f ) ip' must be added to the exponent of P in ( 18 ) . 16 NEIKIKK: groups op OltDEE Jj", WHICH From a consideration of (18), (19) and (20) we arrive at the expression for a power of a general operator of G. (21) [a, y, a;]' = [sz, sy + U,p, sx + F.^"-'] , where V. = {'2){'^+ {a^z + a.^{l)y + cyz + ah{l)z'\p] + a( J ) { 5x2 + ^xy + dyz } xp .* 5. Transformation of the groups. AU groups of this section are given by equations (15), (16), and (17) with a,6,/3,c,, a;(l + op"-') + /9ay"'-'] , (23) [ s , 2/ + ^j^ , a; + ^p^-' ] = [ z , y + 6y> , a; ( 1 + op"-* ) + bx'p"-'^ , (24) [z', y'+^ji), (a;'+<^3p)i)"'-'] = [z', yXl+'^P)' a'(l+dp)^"-»+cxp"'-*], in which ^^ = (?'yz" + 6'xz", e^ = d'y'z", 4>^ = a'xy' + { a'(/3'y' + 6'z')(^) + a'xz + c{yz - y'z) }p, ^ = a'xy" + a':cz" + a'6'(^)z" + c'yz", ^ = c'yz". A comparison of the members of the above equations give six con- gruences between the primed and unprimed constants and the nine indeterminates. 18 NEIKIRK ; (I) (11) (IH) (IV) (V) (VI) tfj = /3y' (mod^), <^j = aa; + /3a;'/) (mod^'), 6^ = by (modp), (f)j^ = ax + bx { mod p ) , 5j= dy' {moip), <^3 = ex + cfe' (mod ^) . The necessary and sufficient condition for the simple isomorphism of the two groups G and G' is, that the above congruences shall be coTisistent and admit of solution for the nine indeterminates, with the condition that X , y and a" be prime to p . For convenience in the discussion of these congruences, the groups are divided into six sets, and each set is subdivided into 16 cases. The group G' is taken from the simplest case, and we associate with this case all cases, which contain a group G, simply isomorphic with G'. Then a single group G, in the selected case, simply isomorphic with G', is chosen as a type. G' is then taken from the simplest of the remaining cases and we pro- ceed as above until all the cases are exhausted. Let K = K^p*", and dv^\^K^,p'\ = 1 (« = a, 5, a, ;8, c, and d). The six sets are given in the table below. I. aj d. 'H d, A D 2 B 1 E 1 1 C 1 F 2 1 CONTArtr CYCLIC SUBGEOtrPS OF ORDEK Jj"^'- 19 The sabdivision into cases and the results are given in Table H. II. Ol b. p. «J A £ c D E i? 1 1 1 1 1 2 1 1 1 ^1 A c. A 3 1 1 1 A c. A 4 1 1 1 A c. A A 6 1 1 1 A c. A A 6 1 1 A ^3 G. c. A A 7 1 1 A ^. c. c. A 8 1 1 A 5. c. c. A A 9 1 1 A ^3 c. A A A 10 1 1 A A c. G. Ac 11 1 1 * A Ax 12 1 A ^3 c. A « A 13 1 A Ao « « Ao Ao 14 1 A Ax c. A Ax Ax 15 1 A A. c; A Ao Ao 16 A Ac » « Ac Ao The groups marked ( * ) divide into two or three parte. Let flwZ— 6c = 5,p*', Old — fic = ^p*' and a^h — a^ = XiP^' ^*^ ^, , ^, , and Xi prime to p . 20 NEIKIBK : GEOTJPS OF OIIDER fl^, WHICH III. * 0, I; X, « e. f. X, ^n 1 A As 1 J^r c„ c. As c. c^ 1 Ci Ae 1 c. On c. -D., c. ^. 1 1 D, En 1 F, C.e 1 G, En E, Cu o. 6. T^pes. The type groups are given by equations (15), (16) and (17) with the values of the constants given in Table IV. IV. a 2> a /3 c d a h a /3 c d A 1 A p B, A 1 p A A 1 A 1 p 1 A p 1 A 1 A p 1 Ao 1 K A 1 p 1 A. 1 1 Ao 1 p K c. V 1 A p 1 1 A (O p 1 A 1 A 1 K=l, and a non-residue ( mod p ) , CONTAIN CYCLIC SUBGROUPS OF ORDER p""'^. 21 The congruences for three of these cases are completely analyzed as illustrations of the methods used. The congruences for this case have the special forms. (I) 6'az' = /3y'(modp), (II) ay = a (mod^), (in) h'xz" = hy ( mod p ) , (IV) aa:y" + ah' ( j ) «" + c'yz" =ax + hx'{ mod p ) , (V) (?=0(modj9), (VI) c'y'z = ex ( mod p ) . Since z is unrestricted (I) gives /S = or ^ (modp). From (II) since y' ^ , a ^ ( mod ^ ) . From (III) since x, y', z" ^ 0, 5 ^ (modp). In (IV) 6^0 and x is contained in this congruence alone, and, there- fore, a may be taken = or ^ ( mod p ) . (V) gives d=0 (mod /)) and (VI), c ^ (mod p) . Elimination of y betveen (III) and (VI) gives h'c'z"^ = 6c ( mod p ) so that he is a quadratic residue or non-residue (mod ^) according as h'c is a residue or non-residue. The types are given by placing a = 0,J = l,a=l,y8=0,c = /e, and d — Q whei'e k has the two values, 1 and a representative non-residue of p . The congruences for this case are (I) d'{yz: -y'z) = fiy'{moip), (II) a[ xy' -f a'xz = a^x + /3x' ( mod p ) , (HI) d'yz" = hy' {mod p), (IV) a'xz" = ax -f 6a;' ( mod ^ ) , (V) d'z" =d {mod p), (IV) cx-f tZx's (mod;)). Since z appears in (I) alone , $ can be either = or ^ ( mod p ) . (II) is linear in z' and, therefore, a = or ^ ( mod p ) , (III) is linear in y and, therefore, 6 = or ^ . 22 KEIKIEK : GEOUPS OF OEDEE p" WHICH Elimination of x and z between (IV), (V), and (VI) gives dd} = d' {ad — he) { mod p ) . Since z" is prime to p, (V) gives d^ (mod^), so that ad—bc^O ( mod p ). We may place b = , a = p, ^ = 0, c = 0, d =1, then a will take the values l,2,3,---,p — 1 giving p — 1 types. A- The congruences for this case are (I) d'{ yz —y'z) = ^y ( mod p ) , (II) ajX+/9x' = (modp), (in) d'yz" =hy' {modi p), (IV) ax + 6a;' = ( mod p ), (V) 4). and (5) [y,xy=lsy,x{s + k{l)yp^-*}-\ Placing s=p and y = 1 in (5) we have, from (2) Choosing X so that x + l=0 (mod^""-^), P = ^j P* is an operator of order p , which will be used in the place of Pj,and^j= {P,P} withPP=l. 3. Determination of H^. We will now use the symbol [a, 6, c, d, e,/, •••] to denote Q^P»P= Q^P'P^-. -. H^ and Q, generate G and all the operations of G are given by [z,y,x2 (a=0,l,2,...,/-l;y=0,l,2,...,;>-l;x=0,l,2,...,7)"'-»-l), since these are ^" in number and are all distinct. There is in G^ a sub- group JB^ of order ^""^ which contains ff^ self-conjugately. JET^ is gener- ated by S^^ and some operator [z, y,x'] oi G. Q[ is then in Jff^ and H^ is the subgroup { Q^, H^ ] . Hence, (6) Q:'>PQf = R^P'^, (7) Q^'P Q'l = P^P'J'-^ . To determine a, and /S we find from (6), (5) and (7) By(l) Q:'^PQf = P, and, therefore, a*" — ft*" L_J;3^0(mod^), 1 1 aj = 1 (mod^"—*), and a^= 1 (modp"-') (m>5). Let Oj = 1 -|- AjP""* and equation (6) is replaced by (8) Q^P Q? = JiPpi+a,P^ , t BUENSIDE, Theory of Qroups, Art. 56, p. 66. 24 NEIKIRK : GEOUPS OF ORDJQB p", WHICH To find a and 5, , we obtain from (7), (8) and (5) [-/, 1, 0,/] = [O, 5?, a^Jp'"-«j. By (1) and (4) and, hence, 6" — 1 5j=l(mod^), a j^-—^ = {mod p) , therefore 6j = 1 . Substituting 6, = 1 and a^= 1 + ajP""* in the congruence deter- mining ffj we obtain (1 + a^p'^~'y = 1 (modp'^^), which gives flj = (mod^). Let (Xj = ap and equations (8) and (7) are now replaced by (9) Qi^Q'i = ^sp»+'*-% (10) Q^pRQ^, = IiP"^. From these we derive (11) [-yp,O,x,yp-\ = [Q,^xy,x + {axy + a0x{l) + ^m)y}p'^*-\, (12) [-j(p,a;, 0,y;?] = [0,a;, aicyp""-*]. A continued use of (4), (11), and (12) yields (13) lzp,y,xy = [szp,sy+ U,, sx + r,p'^-*'] where U, = fiil)xz, r,= (I) {axz + fim)z + kxy + ayz} + m{l)a^z + ha0{i,As-l){2s-l)z'-{l)z}. 4. Determination of G. Since ^ is self-conjugate in (tj we have (14) Q:'PQ, = QfB'F'^, (15) Q-'RQ, = q\''R''P^\ From (14), (15) and (13) [-P, 0, l,iJ] = [X;), /., €^ + ly™-^] CONTAIN CYCLIC STJBGROUPS OF ORDER Jj""'. 25 and by (9) and (1) X^ = (modj)^), ef + vp'"-* + \hp = 1 + ap""-*^ (modp"-'). from which €^ = 1 (mod/)*), and €j = 1 (modjp) (ro>5). Let Cj = 1 + €^p and equation (14) is replaced by (16) Q-'Pq, = Qn>R'P'^"-^. From (15), (16), and (13) r /7p 1 ~\ l-p, 1, 0,p-\=^-^p, d', Kp-*]^ where By (10) d' = \ (mod/)), and rf = 1 and by (1) c^p* = op"-* (mod^°— '). Equation (15) is now replaced by (17) Q-'Bq, = Q\^RP'^\ A combination of (17), (16) and (13) gives [-^,0,l,^]=[{,Ci±^^-fcS^}y,0,(H-e,;,)^]. By (9) |^(i±^^)l:i_VcS^}A;,H (1 + e,py ^ 1 + op"'- (mod^-^), y3 = (mod;?). A reduction of the first congruence gives {l+J,Ppzli^,^ + 7Mi'' ^ {« - aS^}y»- (mod;,"-') and, since Il±^:^^lll^ = 1 (mod »), (e,+ 7A)/=0(mod^»-*) and 26 NEIKIEK : GROtTPS OF ORDER jp", WHICH (18) (e, + 7^)?' = (« + \:)p'"-' (mod^"-'). From (17), (16), (13) and (18) (19) [-y,x,0,y-\ = lcx!/p,x, {exy + ac{l)y}p'^-*2 , (20) [-y,0,x,y] = lx{yy+cB{l)}p,B3^,x{l + e^yp) + ep'^'] where 0={e&c + aS7x + e,(a+^)a;}(|)+^ac{^y(y-l)(2y-l)S» -{l)S} + {ayy + Shy + aBxf + {acB'y+ac8){y)}{l). From (19), (20), (4) and (18) If x be so chosen that h + x=0 (modp""-') Q = Q^P'' is an operator of order p^ which will be used in place Qj andQ''=l. Placing A. = in (18) we get £j/ = (mod^"^*). Let tj = ep"""* and equation (16) is replaced by (21) Q-^P Q=QyrJR^ pi+.i^-» The congruence op"-* = chp^ (mod ^"'-') becomes ^pm-i ^ Q (mod^"^'), and o = (modp). Equations (19) and (20) are replaced by (22) [— 2/, X, 0, y] = [cxyp, x, exyp'"-*] (23) [- y, 0, x,y^ = [{yy + cB(l)}xp, 8xy,x{l + eyp'"-')+ dp'^*] where e = eSxil) + {ccyy + % + acS(*) } (l). A formula for any power of an operation of G is derived from (4), (22) and (23) CONTAIK CYCLIC STJBGEOUPS OF ORDER p"^^. 27 (24) [z, y, x]-= [sz + U,p, sy + F., sx + TT.^"-*] where F. = S(;)a«, W, = {\){ecz+[{a'i-\-h]c){l)z+eyz+hcy'\p]+{\){e'ix+€y+hht]xzp +^cS6{^(s-iy-»}(')a;p+i{Sea;+acS(^)}{^3(s-l)(2s-lK-('>}^. 5. Transformations of the groups. Placing y = 1 and x = — 1 in (22) we obtain (17) in the form A comparison of the generational equations of the present section with those of Section 1, shows that groups, in which 8=0 (mod p ), are simply isomorphic with those in Section 1 , so we need consider only those cases in which S ^ (mod^). All groups of this section are given by (25) (26) G: (27) R-'PR = pi+'s'— , Rp=l, Q^ = l, and ^--^ = 1, {k, y, c, e = 0, 1, 2, ...,p-l; B = l,2,...,p-1; e=0,l,2,...,/-l). Not all these groups, however, are distinct. Suppose that G and G' of the above set are simply isomorphic and that the correspondence is given by Since 2?-= 1, Q''=l, and P^"-' = 1, R^ = 1, Q't = 1 and P'f" The forms of these operators are then P; = Q'R'^P', R[ = Q'^'j-p-v'P'i^- 4^ where dv\_x,p'\ = 1. 28 NEIKIRK: GROUPS OF ORDER p", WHICH Since S is not contained in { P } , and Q'' is not contained in {S, P} Ii[ is not contained in { Pj } , and Q'f is not contained in {Ii[, P[}. Let j?;''=p;''-*. This becomes in terms of Q', R and P' [sVp, sy, sVp-^*] = [0,0, say—*] , and s'y = (mod^), s'z = (mod^). Either y or z' is prime to p or s' may be taken = 1 . Let and in terms of Q, M' and P' \_s"z"p, 0, s'x'f)"'"*] = [sV^, s'y, (s'x + sk )/>"'"*] , from which s"z" = s'z ( mod p ) , and s'y' = ( mod p ) . Eliminating s' we find s"y'z" = (mod 2?), dv[^y'z", pI = 1 or s" may be taken = 1. We have then a", y and x prime to p. Since P, Q and P satisfy equations (25), (26) and (27) JR'^, Q[ and Pj do also. These become in terms of P', Qf and P", [z + *;;>, y, X + e^"^"] = [z, y, x(l + kp'^-*)-] , [2+*>,y + «'xz",x+0;p"-»] = [z+4>,_p,2/ + V,a! + 02i>'"-*], [(z' + «i»;);j, y, e>-*] = [(z' + *3)y , y, e^--*], where *i = — c'yz', 0j = e'ocz ■+■ k'xy — e'y'z , *; = {7V' + c'S'(^)} X + c{yz" - y"z), e; = e'xz" + {{D [a'ry'z" + a'c'8'(r) + ^'^'^''l + S'e'x(n + e' (yz" — y" z) + k' xy" } jp , *3 = 72"+ ^s' + c'By'z, 0j = ex + (7x" + Sx + e'Sy'z)p, *3 = c'yV', @3 = e'y'z", $3 = cz", ©, = ex + ex". CONTAIN CYCLIC SUBGROtrPS OF OEDEE p^^. 29 A comparison of the members of these equations give seven congruences (I) ^; = 0(mod^), (II) @[ = kx{moip), 111) *;= j(modp). (rV) B'xz" = By' {mod p), (V) 0; = 0,(modp^), (VI) % = cz"{moip), (VII) 0; = 03(mod^). The necessary and sufficient condition for the simple isomorphism of G and G' is, that these congruences be consistent and admit of solution for the nine indeterminants with x,y' , and z" prime to p . Let K = K^p", A P ^. 1 P ^. P 1 5. P K A 1 P CO K = 1 , and a non-residne ( modp ) , " = 1,2, ■■■,p-l. A detailed analysis of several cases is given below, as a general illustra- tion of the methods used. A- The special forms of the congruences for this case are (11) e'xz' = Jcx { mod p ) , (in) 72" + 5z' = (mod^), (IV) 8'a»" = Sy' (modp), (V) e'xz" = ex ( mod p ) , (VI) cs"=0(modp), (VII) ca;=0(modp). Congruence (IV) gives B^O (modp), from (II) Jc can be = or ^ (modjj), in gives 7 = or ^ 0, (V) gives e ^ 0, (VI) and (VII) give c = c = ( mod p ) . Elimination of a; , a' and z" between (II), (III) and (V) gives Sk-\- ye = (^ modp ) . If Z; = , then 7 = ( mod^ ) and if ii; ^ 0, then 7^0 (mod^). 32 NEIKIRK : GROUPS OF ORDEB Jj", WHICH A,. The congruences for this case are (H) e'xz' + k'xy' = hx (mod^), (HI) 72"+ 8a' = (modp), (IV) S'ajs" = h)' {moAp), (V) e'xz = ex. (modp), (VI) C8"= (modp), (VII) ex = (modp). Congruence (III) gives 7 = or ^ , (IV) gives 8 ^ , (V) e ^ , (VI) and (VII) give c = e = {moAp). Elimination of x, »', and z" between (H), (III) and (V) gives from which SA + 7€ = ^'Sy' (mod p) 8Z; + 7e^ (modp). If ifc = 0, then 7 ^ 0, and if 7 s then k^d (mod ;?). Both 7 and ;!; can be ^ ( mod p ) provided the above condition is ful- fiUed. A- The congruences for this case are (II) e'xz' — e'y'z = kx (mod p), (HI) (IV) (V) (VI) (VII) exz cz" ^ §p is contained in the subgroup jH, of order p"-*, 5^ = { Q^, P} . 2. Determination of H^. Since {P} is self-conjugate in ^, (1) Qr'P§^ = P'+^--. Denoting QlP''QlP^ ■ • • by the symbol [a, 6, c, (Z, • • •], we have from (1) (2) l-yp,x,yp] = [0,x{l + kyp'^-*)'] {m>4). Eepeated multiplication with (2) gives (3) Ivp, xy = [syp, x{s + i(,')yp"-'}] . 3. Determination of H^. There is a subgroup S; of order p"-' which contains H^ self-conjugately.* H^ is generated by E^ and some operator * BdBNSIDK, Tharry of Groups, Art. 54, p. 64. 34 neikikk: geoups of oedeb ^""j which jBj of G. R\ is contained in H^, in fact in {P}, since if R^ is the first power of jRj in {P}, then H^ = {P, , F}, which case was treated in Sec- tion 1. (4) R\ = P'^. Since H^ is seK-conjugate in H^ (5) Ii-'PR, = Q»/P^K (6) B-^Q^B, = Q\''P'''''. Using the symbol [a,h,c,d, e,f, • • ■ J to denote R'^Q\P'JRfQ\P^ ■■■, we have from (5), (6) and (3) (7) [-p,0,l,p] = lO,^I^p,al + Mp-\, and by (4) a^ = l(modj)), and aj = l(modp). Let flj = 1 + fljp and (5) is now replaced by (8) Ii-'PP, = Q»''P'+''"'. From (6), (8) and (3) 6'— 1 l-p,p,0,p] = [0,b'-p,a,^^—jP+a,Up''}, and by (4) and (2) and therefore 6^ = 1 (mod p), and 6 = 1. Placing 6 = 1 in the above equation the exponent of P takes the form a,y(l + Crp) = a,i (trW ^ from which a^p''{l + U'p) = (modp"-^) or o, = (modp"-') ("»>5). Let fflj = ap"^^ and (6) is replaced by (9) R-'q\R, = q\p-^\ (7) now has the form [-P, 0, l,p] = [0, ySiVp, (1 + a,p)' + J^=], where i\r= p and ilf = /3A I (l±^i4^ - 1 C0NTA.I1T CYCLIC SUBGROUPS OF OEDEE p"'^. 35 from which (1 + a,py + ^^'^Yj~^ ^^P' = ^ {rnodp-') or (1 + a,pY — l ^2 {a,+ ^h}p^^O (mod p-') and since (1 + SP)^-1 '^2?' = 1 (mod^) (10) (a, + /3A)p2 = (mod^"-»). From (8), (9), (10) and (3) (11) [_y, 0, a;, y] = [0, ^xyp, x{l + a^yp) + Op^-'], where 5 = ay9a;(J) + /3^(^)y. By continued use of (11), (12), (2) and (10) (13) [z,yp,x'\'= [sz, (sy4- U,)p,xs+ F.p], where U, = P{;)xz y. = {'^){aiXz+{ayz + hcy + ^k{l)z]p'^-'} + {^{'zW^ + ^«/3 [^8(5 - 1)(2» - l)z» - {;)z]x]r-' Placing in this y = , a = 1 and s=p, determine x so that x + Z= (modp""'*), then R = R^P" is an operator of order p which will be used in the place of ^j , -ff' = 1 . 4. Determination of G . Since H^ is self-conjugate in G (14) Q-'PQ, = R'Q\^P'\ (15) Q-'RQ, = R'qfP'^''. * Terms of the form ( Ai^ -\- Bx) p'"—* in the exponent of P for p = 3 and m > 5 do not alter the result. 36 NEIKIRK : GROUPS OF ORDER p™, WHICH From (15) by (13) and (16) (ej4-<^A)/=0(modp— »). From (14), (15) and (13) (17) [0,-p,l,0,p]=[i;,ilf/>,e^ + iyp]. By(l) €^ = l(modj3), and e^ = l(modp). Let e, = 1 + €jp and (14) is replaced by (18) Q-'FQ, = Ii^Ql''P'+"^. From (15), (18), and (13) [0, -p,o,i,^] = |^c^^£^c^p,^p]• Placing X = 1 and j/ = — 1 in (12) we have (19) [0,-p,0,l,p-\ = [1,0, -ap-*:\, and therefore ^ = e,xy + { e^k(l) + ad{l)y]p'-\ Placing a; = 1 and y = p'va. (23) and by (16) and by (19) a = (modj>). A continued multiplication, witb (11), (22), and (23), gives (Q P^y = Q^P'^ = p^*+')p« . Let a; be so chosen that (cc + ^) = (mody-'), then Q = §j P^ is an operator of order p- which will be used in place of Q^, Q''= = l,and A= (mod ;?"'-=). From (21), (10) and (16) e^p-=lqr~*, a^p^ = a and e^^J^ = (modp'""^). Let Cj = sp""-', flj = ap"*-* and Cj = cp'""'. Then equations (18), (20) and (8) are replaced by (24) (25) (26) 38 NEIKIRK : GROUPS OF ORDER p", WHICH pp = l, Q^' = 1 , P^"'' = 1 • (11), (22) and (23) are replaced by (27) [- y, 0, X, y] = [0, ^xyp, x + p'^-'] , (28) [0, - y, cc, 0, y] = [yxy, e^p,x + ,p''-'2 , (29) [0, -y, 0, X, y-] = [x, dxyp, ^p'^-'], ,^ = axy+/9i(^)y, ^, = ^7(^)2; + % + ^7(l)y, <^, = exy + {eyx{l) + {l){ayy + dyJc{l) + %) + ^yy{l)}p, (^j = cx2/. A formula for a general power of any operator of G is derived from (27) , (28) and (29) (30) [0,z,0,y,0,x]'= 10, sz + U,p,0, sy+r„Q, sx + W,p^~'}, where U, = (0{ 8xz+dyz + ^xy + ^-/{Dz } + \dx{^,s{s-l){'2.s-l)z'-{;)z]x + py{',)a?z, V, = y{\)xz, W,= {l){exz+ [axy + eyz + (/3% + a/37+ 8fe)(^)]p} + (3) {ayx^z + dkxyz + Bhe'z + ^kx-y + 20yk(^l)xz} p + 0yk{l)7^zp + ^{i,s{s-l){2s-l)z'-{l)z}{eyx+dyk[{l)}p + ldyk[l{s-l)z^-z]{l)x\ A comparison of the generational equations of the present section with those of Sections 1 and 2 , shows that, 7 = ( mod p ) gives groups simply isomorphic with those of Section 1 , while /3 = ( mod p ) , groups simply CONTAIN CYCLIC SUBGROUPS OF OEDEB ^'"-'. 39 isomorphic with tiiose of Section 2 and we need consider only the groups in which /3 and 7 are prime to p . 5. Transformation of the groups. All groups of this section are given by equations (24), (25), and (26), where 7, /3 = 1, 2,-.-, p-1; a, S, , s"x>"-*] = [0, s'z'p, (sx + sV)i3'"-'] and s"z" = s'z' (mod^), s"y" = (mod p), and y" is prime to p , since otherwise s" can = 1 . Since E, Q, and P satisfy equations (24), (25) and (26), P, , QJ , and Pj must also satisfy them. These become when reduced in terms of R' , Q' and P [0, z+e'^p, 0,y + y'xz', 0, a; + ^>'"-n = [0, z -f e,j), Q,y + 72/", 0, X + t.P'""']. 40 NEIKIKK : GKOtJPS OF ORDER y", WHICH [0, {z" + e',}p, 0,y", 0, (x" + f,)p'-^*} = [0, (z" + d,)p, 0, y", 0, {x + yfr,)p--n, = [0 , » + 03_p, 0, y, 0, 03 + taP"""']' ^1 = 72" + s»' + <^'7y"2i t; = eW + { e'y'xil') + {l)[a'y'z' + Ve'c^'^'CD + ^^^'^' + ^W\ + ^y{l)z' + e'(y»' — y») + a:xt/'}p, yfr^ = ex + { Sx' + 7x" + e'yy"z } p , 6'^ = d'y"z, 6^ = dz, -f j == e^V, V^j = rfx' + ex, e'^ = Pxy"-d'y"z, e^ = fiz, a/tj = e'xz" — e'y"z + a'xy" + /3'e' ( J ) y", -tlr^^i: ax + ^x. A comparison of the two sides of these equations give seven congruences (I) 0[^e^{moip), (II) y'xz' =yy" (moip), (in) Vr; = V^, (modp^), (IV) ^;^^,(modp), (V) V^; = V^,(modp), (VI) ^; = ^3(mod;5), (VII) ^Ir'^ ^^^ (mod p). (VI) is linear in z provided d' ^ ( mod p ) and z may be so determined that yS = (mod jo) and therefore all groups in vrhich d' ^ (modp) are simply isomorphic vrith groups in Section 2 . Consequently we need only consider groups in which rf = (mod^). As before we take for G' the simplest case and associate with it all simply isomorphic groups G . We then take as G' the simplest case left and proceed as above. CONTAIN CYCLIC SUBGEOUPS OF OBDEE p" 41 Let K = /Cjp*« where , (III) e ^ 0, (IV) and (V) (Z = e = 0, (VI) /3 ^ 0, (VII) is linear in x and a = or ^ ( mod p ) . Elimination of y" and z between (II) and (VI) gives /3 Va3^ = /Sy ( mod p ) and ^7 is a residue or non-residue (modp) according as /S'7' is a residue or non-residue. A3. (I) yS'7'(2)2' + ^xy = yz" + &' (mod _p) , (II) y'xz' = yy" (mod^), (III) e'z = e ( mod p ) , (IV) d=0(modp), (V) e'y"z = ex ( mod p ) , (VI) ^xy" = ^z {moip), (VII) e'aw" — e'y"z + ^'e'(l)y' = (xx + /Sx' (mod p). (I) is linear in z" and S = or ^ 0. (11) gives 7 + , (III) e ^ 0, (V) e ^ and (VI) /3 ^ . (VII) is linear in x' and a = or ^ (modp). Elimination between (II) and (VI) gives /3'7'a^ = y97 ( mod p ) , and between (II), (III), and (IV) gives e' 76 = e^ y'e (mod p ) . /S7 is a residue, or non-residue, according as ^y is or is not, and if 7 and e are fixed, c must take the (p — 1 ) values 1,2,---,^ — 1. CONTAIN CYCLIC SUBGROUPS OF ORDER p^~^- 43 (I) ^7'(j)a' + ^^' = ¥' + &' (mod ;>), (II) ^/xz = 7y" ( mod _p ) , (in) e[xz' + ^'xz{l) = e^x+Scc' + Tx" (modjj), (IV) ex=0(modj9), (VI) |S'ccy"=ySa'(modp), (VII) aaj + y3a;' = (mod p) . (T) gives S = or ^ 0, (II) 7 ^ 0, (III) is linear in x and gives e, = or ^ , (V) e = , (VI) /3 ^ and (VII) is linear in x and gives a = or ^ . Elimination between (II) and (VI) gives y3'7'ar = /87 (mod jj). (I) P'i{l)^' + /3'a^' = 7/S" + &' (mod ^ ), (II) 7'aK' = 7y' ( mod p ) , (IH) €[7Z + eYa3(f ) + ^7(0 + e'(y«' - y'z) = CjCC + Sa;' + tx" + e'+*P"-.t Hence (2) [-y,x,2/] = [0,a;(l + i2/i3"'-O] {">>4). and (3) ly,x-\'=lsy,x{s + 1cy{;)r-']]- Placing y = 1 and s = jo in (3), we have, [QjP^]'= QlP'f = P'^'''* and if x be so chosen that (x-f A) = (modp"'-^), Q = Q^P' will be an operator of order p which will be used in place of 3. Determination of H^ . There is in G' a subgroup H^ of order p" ' , which contains H^ self-conjugately. H^ is generated by H^, and some operator R^oi G. R^^P'". We will now use the symbol [a, 6, c, dl, c, y, •••] to denote R\q'P'R[(^P'--. The operations of H^ are given by [s, y, a;]; (s, y = 0, 1, • • •, p— 1; 03 = , 1 , • • • , p""^— 1 ) . Since H^ is self -conjugate in ^ (4) R-'PR,= q\P-^, (5) 2?r'Q-Si=<2'i'^''^'^'- * BURNSIDE, Theory of Groups, Art. 54, p. 64. tibid., Art. 56, p. 66. where Hence CONTAIN CYCLIC SUBGROUPS OF ORDER p""^^. 45 From (4), (5) and (3) [-i>, 0, l,p] = [o, j5|j-8, «? + Qr-\ = [0,0,1], «?^^«t-i „oK-l J^rzKX (6) °^~ ,^ y3 = (modjp), a\ + ^Ip'"-" = 1 (modjj™-^), and o^ = 1 (mod^"'-^), or a^ = l(modp"'-') (m>6), a, = 1 + ajp*^'. Equation (4) is replaced by (7) R-'PR^=Qi'P^+^^, Prom (5), (7) and (3), [-p,l,0,^] = [0,ij,a|£:ip'-*]. Placing x = lp and y = 1 in (2) we have Q^^P^'Q = P", and 6" — 1 6^=l(modp), a-=r =- = 0(modp). Therefore, 6, = 1 . Substituting 1 for 6j and 1 + a^p""* for a^ in congruence (6) we find (1 + aj)'"-^)" = 1 (mod^"^'), or a^ = (modp). Let a^ = ap and equations (7) and (5) are replaced by (8) -ff r' ^^i = Q^ P'+'^* , (9) Ji-'QP, = QP"^. From (8), (9) and (3) (10) l-y,0, X, 3/] = [0, /3ccy, x+ {oxy + a^x(^) + ^A;y(f)}p"-*], (11) [-y,x,0,yj=[0,a;,axyp"'-*]. From (2), (10), and (11) (12) [z, y, as]' = [sz, sy + CT, sx + F.p'"-*], 46 NEIKIEK : GK0UP8 OF ORDER ^""j WHICH where Placing z = 1 , y = , and s = p in (12) If aj be so chosen that a;+ Z = {modip^-*) then B = H^P" is an operator of order jj which will be used in place of ^j and ^ = 1 . 4. Determination ofG. G is generated by S^ and some operation S^ . ^p = pxp. Denoting S''^E''Q'P^ • • • by the symbol [a, 6, c, tZ, • • •] all the opera- tors of G are given by [t),z,y,a3]; (v, z, y = 0, 1, • • •,p - 1 ; x = 0,l, ■ •■, p'^-^-1). Since j^ is self -con jugate in G (13) xyj-'P*Si = i^Q'P'N (14) S-'QS.^JS'Q'P^"^, (15) /Si P/Sj = A-^ Q^ P^'^. From (13), (14), (15), and (12) l-p, 0, 0, l,p] = [0, L, M, ej + iVp"-*] = [0, 0, 0, 1] and e^ = 1 (modp"-*) or e^ = 1 (mod p""') (m>5). Let Ej = 1 4- e^p'^~^. Equation (13) is now replaced by (16) S-' PS, = i?vg«P+'=^'. If\=0(modp) and \ = \'p, * The terms of the form {Ax + Bx^ )!>'""' which appear in the exponent of P for j) = 3 do not alter the conclnaion for m > 5 . 5 S^P is of order p*"-*- We will take this in place of S^ and assume d'u [X, j)] = 1 . ;Sf -' = 1 . There is in G' a subgroup H\ of order p"^* which contains { /S, } self- con jugately. ^; = {,S'j, /S'^^Q^P^} and the operator T=R'Q>P'' is in H[ . There are two cases for discussion. 1°. Where x is prime to p , T is an operator of H^ of order p""^ and will be taken as P. Then H[={S,,P]. Equation (16) becomes S-'PS^ = P'+"^. There is in ff a subgroup ff'^ of order p""' which contains JI[ self-con- jugately. H',= {ff[,S';R''Q''P"}. T' — P''Q^ is in ffj and also in J7j and is taken as Q, since {P, T'} is of order p"^^. H'^ = {S'x-, O = { -^i^-fi^i } and in this case c may be taken = (modp). 2°. Whtre x= x^p. P' is in { S^) since X is prime to p. In the present case ^' §>' is in ff[ and also in ^j. If z ^ (mod p) take H'Q' as P; if zs (mod^) take it as Q. ff[ = {S„R} or {S,,Q}, and S-'S, R = >^J+*'^-^ or Q-' S,Q= S\+'"'P'-* . On rearranging these take the forms Sz'RS, = RS^'^ = R:P^' or S-^ QS,= QS^^ = QP'^"' , and either c or gf may be taken = ( mod p ) , (17) cgr = (mod p). From (14), (15), (16), (12) and (17) 48 NEIKIRK : GEOTJPS OF OBDEB.-p", WHICH Place x = \p and y = 1 in (12) Q-^P^p Q = P>'P or S\qS\=Q, and (?"= 1 (modjj), d = l. Equation (14) is replaced by (18) S-'qS.^R'QP^'^. From (15), (18), (17), (16) and (12) [_p, 1, 0, 0,^] = [o, /^ ^^^g,wY^\ . Placing X = Xp , y = 1 in (10) and f =1 (mod p) , /= 1 • Equation (15) is replaced by * (19) S-'IiS,= EQ'F^^. From (16), (18), (19) and (12) S^PSl = pi+'=J^ = P and ^2 = (mod p). Let e^ = ej} and (16) is replaced T>y (20) S-'PS, = By Qip^+'p-^. Transforming both sides of (1), (8) and (9) by S^ S-' Q-'S, ■ S-'PS^ ■ S-' QS, = SfP'+'p-^S,, S-'P-'S, ■ S-'PS, ■ S-'PS, = S-' Q» S, ■ S-'P'*'" -^s,, S-'R-'S, ■ Sf QS, ■ S-'ES, = S:' QS, ■ S-'P^'^S,. Reducing these by (18), (19), (20) and (12) and rearranging [0, 7, S + ^c, 1 + { e + ac + i + acS + a^{l) - ayjp'-'-'] = [0,7,S,l + (€ + i);,"-*]. [0,7,^+S, 1+ {kg + e + a + aB - ayg } p^-*] = [0,7 + /3c,/3 + S,l+ {e+a+/3e + a(?)c + ay37}p-'-*], [0, c, 1, (e + a);)"-"] = [0, c, 1, (e + a)p'^-*]. CONTAIN CYCUC StJBGBOUPS OF ORDER p""-^ 49 The first gives (21) i8c= (mod p), (22) ac 4- acS — ay = ( mod p ) . Multiplying this last by g (23) agy = ( mod p ) . From the second equation above (24) gk + aB = fie+ a0y (mod p ) . Multiplying by c (25) acB = (modp). These relations among the constants must he satiKJied in order that our equations should define a group. From (20), (19), (18) and (12) (26) [-y,^,0,x,y] = {0,yxy + x,{^,y), ^y + <^,(a:, ?/), r- + @i(a;, y)K~']' (27) [-y, 0, X, 0, y] = [0, cxy, x, @,{x, y)p'''-*}. (28) [-y,x, 0, 0, y] = [0, x, ^.^y, @,{x, y)p'"-*], where X,(x, y)=c&B(?), «,(x, y) = €xy + (I) [7ix + eSx + aSy + {ay + kS){l)] + (?) [c^' + e^7]a; + it) [«7y + % + a%'] + /^Y-HOy ' ej(x, y) = exy + 9x(^) + ac(Oy, 63(2;, y) =Py + «9^i'i) + 05^(2)2/- Let a general power of any operator be (29) \_v, z, y, x]' = [sv, sz + U,, sy + F, >^x + W,p—*}. 50 NEIKIRK : GROUPS OF OKDER p", WHICH Multiplying both sides by [v, z, y, x] and reducing by (2), (10), (11), (26), (27) and (28), we find U,.,., = U, + {cy + yr)sv + c8(y)a; (mod jj), Z^, = V^ + (gz + Bx)sv + 75r(y )a;+ /e7(j)sr +^{sz+ U,)x{xa.od p), TF,^, = Tr.+ 0,(ai, 6t)+ {ey+jz-\-ayxy + ac{l) + ag[:,)} sv + { ax + ^k{l) + ay + aBax + agsvz} sz + ksxy + {%'){ cjy + egz} +U,{ax+ ^k{l) + ay + a{Sx-h gz)sv} + a${"Y')x-i- kV,x (modjp). From (29) l\ = 0, V, = 0, TT, s 0. (mod p). A continued use of the above congruences give U, = (cy + 'yx){-,)v + -lcSx^;{^(2s _ 1)^- 1 ) (^)(modp), F = { [gz +h.: + 0y{^^)v + ^xz]{;) + ^ygaw { J(2s - 1) t, _ 1 }(J) + ^y{l)arv (modjj),, ^^^. = iDi^'^ + e'/^ + {ay+Bko + /3fe)(^) + ffyk{l)v + a.c{l}v + Jvz + ag {l)v + (xxz + hey + ayxyv + ayz] + (' ){ aca^ + ayx~v 4- 'i^ylc{l)xv + ghxzv + hhx?v + ^Ta^z + acvy^ 2g 1 + ayxvy ] -f ^ky{\)x'v + (;) — g— { 087(^)1;'' + ahxso + agvz-] + lv{\){\{2s — l)v-l}{yjx+ ehx+ahyx + acS(^) + yglc{l) + cjy -)- egz} + !(;){( J)«' - (2s -l)v + 2]{chjx+egyx]v+l{'^){\{s-l)v-\){a.ch + ygk}x'v + la^x{'^){\{2s-\)z-l\z + lahy3^v{l)l{Zs-l){moAj>). CONTAIN CYCLIC SUBGROUPS OF ORDER p^'^. 51 Placing v = \,z = y = s=p\a. (29) [ ^1 P^ ]' = /S^ P^ = PC^+')P (i> > 3 ) ♦. If o; be SO chosen that a; + XsO (mod^"-*). S = S^P'' is an operator of order p and is taken in place of S^ . ^' = 1. The substitution of S for S^ leaves congruence (17) invariant. 5. Transformation of the groups. AU groups of this class are given by (30) with G: 8-'QS=^F'QF^-^, S-'FS^FQ'F^"-^*, {k,^,a,a,y,B,e,c,e,ff,j = 0,l,2,---,p — l). These constants are however subject to conditions (17), (21), (22), (23), (24) and (25). Not all these groups are distinct. Suppose that G and G' of the above set are simply isomorphic and that the correspondence is given by IS, F,Q,F] ^-[5;,p;,q;,p;J- Inspection of (29) gives S[ = s''"'R''"'qy"'F^"''^, R[ = s''"s:'"q'"F'"'^, Q[ = S"'R'--'Q''''F^'f'^\ F[ = S'^R'qyF\ * For jp = 3 and ei^yg= Py = (mod p ) there are terms o£ the form {A-\- Bx+ Cx' + Dx' jp"*"* in the exponent of P. For m > 5 these do not vitiate our conclusion. For ^ = 3 and d, yg, or /3y prime to p, [Sii^]^ is not contained in {P} and the groups defined belong to Class II. 52 NEIKIKK : GEOXJPS OF ORDER p^, WHICH in which x and one out of each of the sets v ,z' ,y' ,x' ; v" , «", y" , x" ; v'", z"\ y"\ x" ; are prime to p . Since S, JR, Q, and P satisfy equations (30), 8[, Ii[, Q[, and P[ also satisfy them. Substituting these operators and reducing in terms of 8' , If ., Q , and P' we get the six equations (31) [f:,^:, r-:,x:]=[K,2r., r.,x.] (K=i,2,3,4,5,6), which give the following twenty-four congruences (32) where V:= F. (niodp), Z^ = Z. (modp), r:^f;(modp), [X>X,(modp-»), Z, = z, f; = v, r, = v, Z[ = z + c'{yv — y'v) + y'xv' + cBx{l'), X; = 35 + {e'a3«' + (yJx + e'B'x -f- a'S'7'x)('') + c'S>"(J') + (ay'v'+S^k'v' + a'S-^V + ^'k'z){l) +JXzv -zv) + e'g'lzd') - z'(j)] + <'VL{l>' + i2>-'^'''^] + e(yv'-yv) + cj'[y(l') -y'{l)-] + a'c'[(^)t)' + v{-l') — yy'v] + a'{yz — y'z) — a'^xz'^ + a'xz + «'^a3(f ) -f aVa3(y — y')«' -f AV}j''^S ^2 = ^. F2 = 'y + ySr', Z; = z + c'( yv" - y"t) ) + yW + e'S'( •" ) , ^2 = 2+ /Sz' + c'ySy'u , ^2 = y + g'i^'"" - 2"«) + 8'»u" + 7ya!(r) + /3V(^)u" + ^xz", CONTAIN CYCLIC SUBGROUPS OF OBDER jo"-'. 53 X', = x+{@[{x,v'')+JXzv''-z"v) + eg[z{l'')-z''{l)-]+ag'lil)v' + (-l")v - zz"v-\ + e{yv"-y"v) + cj[yil")-y"{l)l+ «'c'[(|)^" ( 2)'" — yy^'J+rtS'(»'y —zv)z + a[yz —yz)+aovz + a'yXy - y")v"x + axz" + a'ffxd") + ^h'{l)z' + k'xy"}p'"-*, X^ = x+ {ax + fix' + d{l)y'z + e'^vy + (cj^ + eg'^z'){l) + a'cXOi')v+)^vz' + agX^^") + a'^{gzv + y')z}p^-\ ^ = z +c{yv —yv), Z^=z, T^ = y +g{zv -zv), Y^ = y, ^; = {^' +/(«V' - z"v) + egi{t')z- (r)s"] + ag'l{'^)v"+ {-'■')v -8VV]+e'(2/V'-2/V)+cJ[3/'(r)-2'''(r)] + «c[(0«''+(T)^' — yyo ]+a(ya —y z))f^\ y\ = v, F^ = v + 7^" + &j', Z; = a + c'(t/r"'- y'"^) + I'xv!"^ c'S'xil'"), Z^ = z + '■fz"+ Sz'+ c'[{l)v"y"+ i\)'"'y'+ '^'{"ty" + V)" + c'7%"«> 1": = y + f7'(2^"'- 2'"^) + ^»^""'+ •y'y'^cr") + ^i{%v + /3W", x: =x+ {0; (x, ^,"') +/(zv"'-/"^)) + ey [(r)» ^ (2)3"1 + ay [(;K' + {-^")v - z:^"v ] + e{yv"' - y'"^) + cj\_y{f) - /'(J)] + ac'\_{l)v" + {-f)v- yy"'v"] + dgXv"'z-m")z" + aXyz" - y '») + a'SWV" + aYa;(y - y"')«"' + «'a»"' + a'/9'x(r ) + ^Tc'z"{l) + ^V"} ;>"-'• 54 NEIKIRK : GROUPS OF UBDEB p", WHICH X^ = x+ {ex+Sx +yx"+{l) [a'c ( f ) v" + dy"z + ev"y" + j VV + a^(i )v +{cjvy +egvz ){v + 6v) + a{z + 6z)v z + ?^ a'srVV^ + i [^(27 - l)v' - 1] (c'jV" + e'g'z")v"-\ + (?)a'cVy' + («) [aV(f )«' + a'y'z + eVy' +/t;V + '^'ffXl')'"' +/cWy' + e'g'vv'z' + dg'v'zz -\- a'c'yy'y'v' + —3— a>rV' + 1 { ^ (2S - 1 ) z,' - 1 } ( c'jy' + e'^V)] + («)a'cV/+ (r + 8«')[/7«" + (t')ay + eVy" + (T )«'«=' + a'g\z + Sa')] + {'\'" )[e'g'yz" + c'/yy"] + S[(e>V + c'jV')(;) + e'ry' +/2' + a'zy + a>'^z»' + a'yz'y" + dc'yvy'y"] ^5 = "'. ^5 = ''' + ct;"' -^.^ = 2' + c' ( y V" — y"V ) , Zj = s' + ca" + c'cy"» , ^5 = y' + S''( 2'«"' — z'V) , r^ = y' + cy" + g'cv'z", ^.; = {^' +/(«V" - z"v) + eg'[{l"')z - (O^'"] + agl{-)v" + (-='") «' - ^VV] + e'(yV" - y"v) + cj'[y'{^;') - y"'(J')] + a'c'[(Oz,"' + (-f )«' _ yyV] + a'(yV" - y'V)}^'"-*, X,= {x'+ex + cx +d{l) y"z' + jcv'z'+ ( eg'cz+ c'cj'y" ) ( ^ + e'cy'v + a'cy'z + dg'z'v + ay(=f ) + a'c'(=f ), ^6 = «"> ^6 = ''" + S-^'. , = s + c(3/ z; - y t, ), ^6 = s" + gz, ^6 = y" + g'{ 2 V" - z" V ) , r^ = y" + gy\ ^; = {a;" +j\z"v" - z"v") + ey[(J"'y' - (i"y'] + ay[(^")«"' + (-fV - «V"t;"] + e'(yV" - y"V) + c'j'lyXl'") - y"'(r')] + o'c'[(f ).'" + (-f )z;" - y"y"V'] + d{y"z" - y"z))f^, ■^6 = { a;" + ;x + gx + a'gryV } f-^. CONTAIN CYCLIC SUBGROUPS OF OEDEE p"^^. 55 The necessary and sufGcient condition for the simple ismorphism of the two groups G and G' is that congruences (32) shall he consistent and admit of solution subject to conditions derived below. 6. Conditions of transformation. Since Q is not contained in { -P } , R is not contained in { Q, P } , and S is not contained in {R, Q, P] , then Q[ is not contained in {P[} , R[ is not contained \a {Q[, P[} , and S[ is not contained in {R[i Q[, P[} . Let q['' = P["^. This equation becomes in terms oi S' , R , Q' and P [sV, s'z' + c\',')v'y\ s'y' + g'{;')vz'. Dp-*] = [0,0,0, say-^], and s'v = s'z = s'y = (mod^). At least one of the three quantities v, z' or y is prime to p, since other- wise s may be taken = 1 . Let R[-" = Q[-'P["^, or in terms of S', R , Q' and P [sV, s'z' + c'('")«Y', «Y' + ^'(5")""^% Ep-'] = [s'v', s'z + c'('')«y . «y + ^'(r)'''2'' E,p-'-\, and sV'= s'v' (modp), s"z" + c'(f )«"/ =«'«' + c'(0«y (mod;)), Since c'jr' = (mod p), suppose gf' = (mod p). Elimination of s between the last two give by means of the congruence Z'^ = Z^ (mod p), s" { 2 {y'z" - y"z) + cy'y" {y - v") } = (mod ^i*), between the first two s" { 2 {v'z" — v"z') -f c'vv"{y' — y") } = (mod jj), and between the first and last s" ( y'v" — y"v' ) = ( mod p ) . 56 NEIKIBK : GROUPS OF OBDEB p" WHICH At least one of the three ahove coefficients of s" is prime to p , since otherwise s" may be taken = 1 . Let S['"' = R['"Q[''P['^* or, in terms of S', If, Q', and P' = [sV + sV, «"z" + sV + c'{{'^')v"y" + {'^)v'y' + ss'y'v ] , s'Y + s'y + g' {{•;)v"z + ( J')«V + s's'v'z" ] , ^jp"""] and s V ^ s V + SV (^ mod p ) , s"z'+c\^")v"Y=s"z+s'z'+ c'{{-;')v"y"+ (5')^y+ «'« VV } (modi?), s"y"+?'(;"')«"'2"= av'+ «y+ y{(r')''"«"+ (r)^«'+ ««"'"'''} (modi>). If jf' s and c' ^ (modp) the congruence Zj s Zj (modp) ^ves (yV — yV) = (modp). Elimination in this case of s" between the first and last congruences gives s (jy V — y») = (modpj. Elimination of s" between the first and second, and between the second and third, followed by elimination of s' between the two results, gives s Iz — cy z V + j-y v \ [yv — y v ) = v (modp). Either (y"v"' — y"'v"), or {y'v" — y"'v') is prime to p, since otherwise s" may be taken = 1 . A similar set of conditions holds for c' = and ^' ^ (modj)). When c' = ^' = (modp) elimination of s' and s" between the three congruences gives / ff ff- V V V fff A " s A = s y y y f ft ff z z z = (modp) and A is prime to p , since otherwise s" may be taken = 1 . 7. Reduction to types. In the discussion of congruences (32), the group G' is taken from the simplest case and we associate with it all simply isomorphic groups G. CONTAIN CYCLIC SUBGROUPS OP OBDEE ^'"~'. I. 57 A • B. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 a, 1 J 1 1 1 1 1 1 0~ 0~ f2 1 1 Si ± 1 >J 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 s, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 58 NEIKIBK : GEOTJPS OF OKDEB Jj", WHICH II. A. T" 2 1 >< ' X ' 2 ><" ' ^1 3 X ' 4 5 9^' s 7 9'6' s 9 9i" V! 11 12 Xl9l 13 X 14 19 ^1 IS [9^ r ^1 18 i5[ 19 x" 20 21 IT, 22 19^ 23 i9i 24 25 "1 26 i9i 27 i9i 28 2» i9i '1 96' 96 96=-'*2| X »2 "»2 X 192 X "2 X X *2 3 h ^1 ^1 96 '96 '96 '1 92 =•1 "2 2l2 '92 13j 2I2 192 ^2 ^^2 ^ij "2 «2 "2 4 h X X ^6 96 '96 "2 '92 "1 2^2 19, 192 13, 19, "2 192 19, '9l 192 "1 "1 19, 19j "1 "1 5 T ^1 * '96 '96 '96 19, '92 19l 19, 196 "6 ^1 -^16 196 ="1 ^'6 h \ '^6 '96 '96 X '92 X "6 "2 192 "6 "2 X 196 X ae 19, ^1 "6 19, "2 'l "1 7 ~8 h h h '96 '96 '96 2Sj »2 25, "2 »2 2*2 * h "6 '96 196 252 '»2 «2 "2 192 192 212 2l2 24^ i^a 252 9 10 h h * '96 '96196 196 196 2l6 196 "6 192 19? «2 2I2 "6 h "1 h '96 '96 196 a 10 X 26 10 13 10 190 196 «'6 2l6 »4 X 2I2 15^ »4 252 ^ 11 h \ h "6 "6 '96 24j 192 "1 4 «2 192 13, 2I2 192 »2 ■■^ .12 h \ ♦ "6 '96 196 196 25 10 '4 2l6 196 34 «« 13 h \ h >»6 »96 19« "6 • »6 "6 • '92 «6 • 192 h "6 2I2 19j h f*'2 14 h \ * >96 '96 196 "2 • 19, 19, 1*6 "6 h "6 ife h »'6 IS h H h '96 '96 196 "6 '92 1^6 "6 «'2 19.3 '»6 2I2 196 196 ^'6 2l6 192 h "6 19, • h 19, n 16 h \ « '96 '96 196 196 196 "'« 2'6 196 h ^1 "6 196 • «l6 17 H H h '96 "6 196 »2 13, «2 132 «52 »2 * 18 h \ * '»« '96 "6 "6 '96 "6 196 2'6 2» 10 »4 - 34 19 h \ h >9« 196 196 10 13 10 ■2b 10 13 10 2~6 34 26 10 "4 20 h \ « "6 '96 '96 196 196 ''le 196 "2 192 21j 21, "6 =>. ^1 21 h * * '96 "e '9fi ■lb ic 13 10 26 30 13 10 196 '96 2'6 2l6 h 26 10 h 22 \ H * "« '96 '96 l9e '96 2l6 196 «6 '96 196 2l6 2'6 h 34 * 23 h « ♦ '9« '96 '96 196 26 10 34 "6 196 h «6 ii 2i \ h 19, 19, '9e "6 "2 1»6 I't • « "6 « 192 h "6 "2 192 h 2l2 26 ^4 \ * '9« 19, 196 «'6 2'6 '96 h »4 2'6 196 * "6 26 h 24 • 19( 19, 19, '96 '96 2l6 "6 196 h S) "6 196 * «6 27 h \ « * * J9, 19, 19, 19, — 19, 19, 19, 25 1( 19, Is" IC 25" 10 2'6 196 2'6 26 10 h 34 13 10 34 26 10 34 29 • « * 19 "e 19, 19, 2'6 196 2'6 • « • 30 =^4 • * 19 19, ,'9( ! 19, 19, "6 196 »'e '96 19« *'« "« 34 '4 * 31 24 « ■# .9, > 19, i'9. 19« »4 »4 «6 196 • "8 32 • * • 19 5>9, i 19, i 19, 19, «6 '96 2'« « • ♦ For convenience the groups are divided into cases. The double Table I gives all cases consistent with congruences (17), (21), (23) and (25). The results of the discussion are given in Table 11. The cases in Table II left blank are inconsistent with congruences (22) and (24), and therefore have no groups corresponding to them. Let /c = /c,p*» where dv[K^, p^ = 1 (« = a, /3,c, g, 7, d, k, a, e, e,j). In explanation of Table II the groups in eases marked [^ are simply isomorphic with groups in A^ B^ . CONTAIN CYCLIC SUBGEOTJPS OF ORDEB j9"'-^ 59 The group G' is taken from the cases marked \x\. The types are also selected from these cases. The cases marked [T] divide into two or more parts. Let aS(a-e) + 27^ = 73, ttS-/3€ = 7„ The parts into which these groups divide, and the cases vKth which they are simply isomorphic, are given in Table III. III. A^s* dvlli,p] =p 2 J dvll„p} = l h A,B* dv[I^,p] =p 3 J dvlli,p] =1 h A,B* dv[l3,p] =p 3 J dv[I^,p] = 1 h A^B^ dv\_I^,p} =p 19 J dv[I^,p'} =1 19, 2*2 A^-Bn dv{I^,p'] =p 11 J dv[I„p} =1 As.i»B* dv{Ii,p'] =p 19 J dv[I,,p-}=l 2^2 -4„-B^ dv\I^,I^,p'\ =p 19 J dv[I„I„p-]=l 192 1^2 h A^Bu dv\_lT,p'] =p 19 J £?U[77,J3] = 1 A.,^B* dvlIi,p-\ =p 3 J (fv[7,,p] = 1 ^27^15 dv[It,p'] =p ^^l\dv[I„p}=l IS -^^29 -^7, 17 d'y[7,o,jD] =p 2*2 dv[I,„p] = 1 202 A33 X»],^ 21 dvlla,p] =p lie (Zv[73,;>] =1 h -^29 -^22, it, 3(1, 31 dv[I,,p} =p 2^10 dv[I„p-]=l h -4 28 ^2,_ ,2 dv[I^, Z,,^] =p lie \_I„p']=p, [I„p]=l h -^29 -^29, S2 [7„^]=1, [I„p-\=p 25l0 [7„ ;,]=!, [73,p] = l h 8. Types. The types for this class are given by equations (30) where the constants have the values given in Table IV. 60 NEIKIBK : GEOTTPS OF ORDER p", WHICH IV. a 1 1 c 9 Y 6 k u e e j 1 J '0 ooooooooo 1 2 ^ 1 3 , 11 ^ /e 1 1 oooooooo 1 1 * 13 J 1 1 19 1 „ 1 # 1^2 1 1 1^2 1 ♦ 2^2 1 1 2^2 1 24 ^4 1 1 K 1 1 1 ^^^6 1 1 « 1^6 1 1 1 1 1»6 * 2^6 i^io 2^10 1 * 1 K 1 'c = l, and a non-residne (mod;). *' For p = 3 these groupe are isomorphic with groups in Class II. CONTAIN CYCLIC SUBGEOUPS OP ORDER jf 61 A detailed analysis of congruences (32) for several cases is given below as a general illustration of the methods used. The special forms of the congruences for this case are (II) /S'iKs'sO (modp), (III) a'( yz —y'z)='kx[ mod p ) , (IV), (V), (VI) /8«' = 0, /Sz' = 0, /32/' = /S'xz"(modp), (VII) a'( yz" - y"z ) + a'^x ( =" ) = aa; + ySx' + a'^y'z ( mod p ) , (X) a'( y'z" — y"z ) = aa; ( mod p ) , (XI) 7d" + Si;' = ( mod p ) , (XII) 7a"+8z' = (modj9), (XIII) 72/" -f Sy' = yS'ax;" ( mod p ) , (XIV) a'( yz'"— y"'z ) + a'/3'a; (='") = ea; + 703" + &c -f a%'z + aV/z -f a'(|)y"z" (mod jo). (XV), (XVI), (XVII) ct" = , cz" = , cy ' = ( mod _p ) , (XVIII) a'( y'z"— y'z ) = ex ( mod ^ ) , (XIX), (XX), (XXI) 9c' = 0, ^2=0, ^2/' = 0(mod^), (XXII) a'( y'z"— y'"z' ) s^'a; ( mod j> ) , From (II) z' = (mod^). The conditions of isomorphism give A = y y y tl nt ^ (mod jj). Multiply (IV), (V), (VI) by 7 and reduce by (XII), ^7?;' = 0, /37z' = 0, /S7y' = ( mod p ) . Since A ^ ( mod p ) , one at least of the quantities, r', 2' or y' is ^0 (mod/j) and /37= (modp). 62 NEIKIKK : GROUPS OF OEDEB p" WHICH From (XV), (XVI) and (XVII) c s (mod ^), and from (XIX), (XX) and (XXI) gr = (mod p) . From (IV), (V), (VI) and (X) if a = , then /3 = and if a ^ , then ^^ (modp). At least one of the three quantitis /3, 7 or S is ^0 (mod ^) and one, at least, of a, e or j is ^0 (mod^). J.3 : Since z" = ( mod p ) , (XVIII) gives e = . Elimination between (III), (X), (XIV) and (XXII) gives ae — ifcj = ( mod ^ ) . Elimination between (VI) and (X) gives a'/S'g"^ = a/S (modp) and a/3 is a quadratic residue or non-residue according as d^' is or is not, and there are two types for this case. A^: Since y and 2'" are ^ (mod p), e ^ (mod p). Elimination between (VI), (X), (XIH) and (XVIII) gives aS - /Se = (mod^). This is a special form of (24). Elimination between (IH), (VII), (X), (XIU), (XIV), (XVHI) and (XXII) gives Ijk + a8(a — e) -f 2(ae— ae) = (mod p^. A^: Since from (XI), (XII) and (XIII) y" and z" ^ (mod ^), and z" = t;" = (mod jp), (XXII) gives ^' ^ (mod ^). EUmination between (III), (X), (XVIII) and (XXII) gives ae—jk = (mod^). A^: (XI), (XII) and (XIII) give »' = a' = and y', 2'" + (mod ^) and this with (XVIII) gives e ^ . EHmination between (III), (VII), (XVIII) and (XXII) gives ae —jk = (mod jo). A^: Since a = then e or J ^ (mod^). Elimination between (III), (VII), (XVIII) and (XXII) gives ae — ^'i = ( mod p ) . Multiply (XIII) by dz" and reduce Se + '^- = a'/3'2"''^0(modj3). COOTAIN CYCLIC SUBGROUPS OF ORDER p^-^. 63 The special forms of the congruences for this case are W ^a»' = (modp), (HI) fe=0(mod^), (IV), (V), (VI) /St,' = /S«' = , /3y' = ^xs" , (^^11) aa;4-/Sa;'=0 (modp), (^) ax = (mod p ) , (XI) yv" + Bv = (modp), (Xn) 78" + S2 = (modjp), (XIU) yy" + Sy = yS'xz'" ( mod ;) ) , (XIV) Gc + yx" + Sx' = (mod j9 ) , (XV), (XVI), (XVII) cv" = cz" = cy" ^ ( mod p ) , (XVIII) cx = 0(modp). (XIX), (XX), (XXI) gv s gz ^gy^Q(modip), (XXII) > = 0(modjD). (II) gives z' = 0, (III) gives ^ = 0, (X) gives a = 0, (XV), (XVI), (XVII) give c = (A ^ 0), (XVIII) gives e = 0, (XIX), (XX), (XXI) give 5^ = , (XXII) gives j = 0. One of the two quantities z" or z" ^ ( mod p ) , and by (VI) and (XIU) one of the three quantities /9, 7 or S is ^ 0. ^j, : (XIV) gives e = (mod p). Multiplying (IV), (V), (VI) by 7 gives, by (XII), fiyv = yS7z' = fiyy = (modp), and /S7 = (mod jo). A^^: Elimination between (VII) and (XIV) gives aS— ^SesO (mod^p). ^24= (VII) gives a = (modp), (XIV) e = or + (modp). A^: (VII) gives a = (mod jd), (XIV) e = or + (moip). A^i (VII) gives = (mod^), (XIV) « = or ^ (modj?). The special forms of the congruences for this case are (I) c'(2/u' — y't;) = (mod^), 64 NEIKIRK : GBOUPS OF OBDER p"", WHICH (III) ]cx = (modp), (IV), (V), (VI) ^i' = 0, ^z = c'{yv"-y"v), 0y' = O{moip), (VII) ax+ 0x' =0 (mod;? ) , (VIII) c'{y'v"- y"v' ) = ( mod j9 ) , (X) ax = (mod^), (XI) yv" + 8i;' = ( mod p ) , (XII) yz'^+Bz' + c'ySy"v+c'{l)v'y' + c'{y)v"y" = c'{yv"'—y"'v) (moAp), (Xni) yy" + Sy ' = ( mod p ) , (XIV) ex + yx" + Bx' = (modp), (XV), (XVI), (XVII) cv"=0, cz"=cXy'v"-y"'v), cy"=0{modp), (XVIII) ex + cx" = 0{modp), (XIX), (XX), (XXI) gv = Q, gz = c'(y"v"'-y"'v"), gy = 0{modp), (XXII) jx + gx' = (mod^). (HI) gives k = 0, (X) gives a = . Since d» [{y'v"'-y"'v'),{y"v"'-y"'v"),p]=lthendv [c,g,p'\=l. If c ^ 0, r" = y" = (mod^) and therefore gr = (mod j») and if g ^0, then c = (mod p) . ^,j: (XVIII) gives e = (mod;?). Elimination between (VII) and (XXII) gives ag — ^j = (modp), (XIV) gives € = (moAp). ^,j : (XVIII) gives c = ( mod p ) . Elimination between (VII) and (XXII) gives ag — ^j = (modp), (XIV) gives e = or ^ (modp). A^^^. (XVni) gives e = 0. EUmination between (XIV) and (XXII) gives eg — Bj = (^ mod p ) , between (VII) and (XIV) gives aS — /Se = . J,3: (XVIII) gives e = 0. Elimination between (VII) and (XXII) gives ag — 0j = (modp), (XIV) gives e = or ^ (modp). A^,: (VII) gives a = (modp), (XIV) gives e = 0, (XXII) gives j=0 (modp), (XVIII) gives e = ot ^0 (modp). A^ : (VII) gives o = , (XXII) gives j = 0. Elimination between (XIV) and (XVIH) gives ec - 67 = (modp). A,, : (VII) gives a = 0, (XIV) gives e = or + (modp), (XVUI) gives e = 0, or ^0, and (XXII) gives,; = (modp). COKTAIN CYCLIC SUBGROUPS OF OBDER ^"~'. 65 A^: (VII) gives a = 0, (XIV) gives e^O or +0, (XVIII) gives e = or ^ 0, (XXII) gives ^" = (mod p). A^: (VII) gives a = 0, (XIV) gives € = 0, (XVIII) gives e = 0, (XXII) gives^ = or ^ (mod ja). A^ : (VII) a = 0, (XIV) e = or + 0, (XVHI) e = 0, (XXH) j = or ^ (modp). A^: (VII) a = 0, (XIV) e = or +0, (XVIII) e = 0, (XXU) ^' = or ^ (mod p ) . Elimination between (XIV) and (XXII) ^ves eg — S; = (mod p ) . A„: (VII) a=0, (XIV) € = or +0, (XVIII) e = 0, (XXII) ^" = or ^ (mod p ) .