fyxull ^nivmii^ pibmg .AvV'i:vv^H \.^..:r^.s 8817 The date shows when this volume was taken. All books not in use for instruction or re- search are limited to four weeks to all bor- rowers. Periodicals of a gen- eral character should be returned as soon as possible ; when needed beyond two weeks a special request should be ma(Je. Limited borrowers are allowed five vol- umes for two weeks, with renewal privi- leges, when a book is not needed by others. Books not needed during recess periods should be returned to the library, or arrange- ments made for their return during borrow- er's absence, if wanted. Books needed by more than one person are placed on the re- serve list. ON IMPRIMITIVE SUBSTITUTION GROUPS A THESIS PRESENTED TO THE UNIVERSITY FACULTY OF CORNELL UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY BY HARRY WALDO KUHN BALTIMORE Z^t Boti ^attimott ^xtee THE FRIEDENWALD COMPANY 1904 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/cu31924032189684 ON IMPRIMITIVE SUBSTITUTION GROUPS A THESIS PRESENTED TO THE UNIVERSITY FACULTY OF CORNELL UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY BY HARRY ^ALDO KUHN BALTIMORE THE FRIEDENWALD COMPANY 1904 On Iinprimitive Substitution Groups. By Harry Waldo Kuhn. Introduction. The study of substitution groups first arose in connection with the solution of algebraic equations. In tbe earliest work that devotes considerable attention to these groups (RufiQni, Teoria generale delle equazioni, Bologna, 1799), we find the non-cyclic groups divided into three classes which correspond to intransitive, primitive and imprimitive groups. Of the substitution groups considered in this work, the group of the order 8 and degree 4 is the only imprimitive group that receives any attention. In a memoir two years later,* Ruffini shows that the group of an irreducible equation is transitive. When the group of an equation in X is not primitive, Jordan has provedf that the equation is the result of the elimination of y from two irreducible equations of the form Z/^ + aiy™"' +....+ a™ =0, 03" +h{y)x''-^+ .... +6„(2/) = 0. and conversely. The systems of imprimitivity of any imprimitive group G are permuted by its substitutions according to a transitive group P that has a l,a isomorphism to G, and whose degree equals the number of systems in the given set. The invariant subgroup of G that corresponds to identity in P is intransitive and its substitutions leave the given systems unchanged. J It is called the head of G and its order may equal unity. || If the group P is itself imprimitive, the correspond- ing systems can be united into larger ones which are permuted by the substitu- * Memoire della societa italiana delle scienze, Vol. 9, pp. 144-526. Modena, 1801. f Traits des Substitutions, p. 259. t Jordan, loc. cit., p. 41 ; ibid., p. 899. II Dyck, Mathematische Annalen, Vol. 22 (1883), pp. 94, 108; of. also Miller, Bulletin of American . Mathematical Society, Vol. 1 (1894), p. 257. 46 KuHN : On Imprimitive Substitution Groups. tions of G according to a primitive group.* The degree of any solvable primi- tive group is the power of a prime.f In order that an equation can be solved by- radicals, its group must be solvable. It follows directly that any imprimitive group that belongs to a solvable equation has at least one set of p" systems of imprimitivity when p is a. prime number. If a given imprimitive group contains two distinct sets of systems of imprim- itivity, then under certain conditions new sets of systems can be formed from these. This can always be done in case some system of the one set has more than one element in common with some system of the other set.J It is not true in general, however, that a new set of systems can be formed by combining all the systems of one set that have any elements in common with a given system of the other set. In his Trait6 des Substitutions, p. 34, Jordan states a theorem that says this can be done, but he afterwards notes the error himself. || Start- ing with this theorem he proves some very interesting results in reference to what he terms "Facteurs de Non-Primitivitat." An interesting problem pre- sents itself here in the discussion of the imprimitive groups for which the theo- rem is true. An important property of such groups has recently been proved by Maillet.§ The important problem of determining when a given group can be repre- sented as a transitive group of a giyen degree (or in particular as an imprimitive group) has been completely solved by Dyck.^ When the properties of the given group are known his investigations give all the ways in which such a represen- tation can take place. They do not determine, however, how many of the differ- ent representations of the same group are distinct as substitution groups.** This question finds its answer in a theorem due to Miller.ff In the particular case when the degree equals the order there is just one such group. * Jordan, loc. cit., p. 899. t Galois, Oeuvres Mathematiques, p. 27. Cf. also Jordan, 1. c, p. 398. X Jordan, loo. cit., p. 34. II Giornale di Matematiche, Vol. 10 (1872), p. 116. § Bulletin de la Sooiete Mathematique de France, Vol. 88 (1900), p. 15. U Mathematische Annalen, Vol. 22 (1883), p. 94. ** Burnside, Messenger of Mathematics, Vol. 23 (1898), p. 103. tt Bulletin of the American Mathematical Society, 2d Series, Vol. 8 (1896), p. 215. Of. also Giornale di Matematiche, Vol. 38 (1900), pp. 1-9. Kuhn: On Itnprimifive Substitution Groups. 47 The investigations of Dyck just referred to determine also the different sets of systems of imprimitivity which are admitted by a given imprimitive group. In particular, when the group is regular, the number of such sets is shown to be equal to the number of subgroups (not counting identity) that are contained in the group. It is clear that any substitution that is commutative with all the substitutions of a given imprimitive group determines systems of imprimitivity of the group. The number of such substitutions for any regular group is well known to be equal to the order of the group.* The problem that has received the most attention recently in the study of imprimitive groups relates to the construction of such groups. The enumeration of the imprimitive groups of a given degree has been carried through degree fourteen. The methods used in forming these lists have been chieflyof a tenta- tive nature. Recently, however, some theorems have been established that are useful in the determination of imprimitive groups of certain kinds.f Any regular group of composite order is imprimitive. The determination of the number of distinct groups of a given order has been studied from the point of view of abstract groups and from that of substitution groups. By means of the latter method the regular groups whose order is less than 48 have been constructed.^ Some imprimitive groups which do not belong to either of the two classes just mentioned have also been enumerated. These include certain groups whose orders are of a particular form. The orders that have been considered are 1) p.q.yW; 2)^3§; and 3) 8j3^, when p, g- and y are distinct prime numbers. Another important class of transitive groups that has been studied is formed by the groups which are isomorphic to the symmetric and the alternating groups of a given degree.** The necessary and suflBcient condition that a group is mul- tiply isomorphic to a non-regular transitive group has also been determined.ff * Jordan, Journal de l':ftcolePolytechnique, Vol. 22 (1861), p. 153. t For full references to those through degree 10 cf . Miller, Bulletin of the American Mathematical Society, Vol. 2 (1895), pp. 138-145. Those of degree 12 and 14 are determined by Miller, Quarterly Journal of Mathematics, Vol. 28 (1895), p. 193, and Vol. 29 (1897), p. 234. Cf. also American Journal of Mathematics, Vol. 21 (1899), p. 387. J Ibid., Quarterly Journal of Mathematics, Vol. 28 (1895), p. 232. II Ibid., Bulletin of the American Mathematical Society, Vol. 2 (1886), pp. 213-322. § Ibid., Annals of Mathematics, Vol. 10 (1896), pp. 156-8. If Ibid., Philosophical Magazine (5), 43 (1896), pp. 117-135 ; cf. Oayley. ** MalUet, Journal de Mathematiques, 5 serie. Vol. 1 (1895), pp. 5-34. tt Miller, Giornale di Matematiche, Vol. 38 (1900), p. 8. 48 KuHN : On Imprimitive Substitution Groups. In the preparation of the following paper I am indebted to Dr. Miller for helpful suggestions and criticisms. The first section of the paper relates to the imprimitive groups whose ele- ments can be divided into systems of imprimitivity in more than one way and whose substitutions permute all the sets of systems according to primitive groups. A few properties of the heads of such groups are first given. These are followed by the study of the groups that contain a given number of heads. Those that contain more than two heads, all different from identity, receive the most atten- tion. The cases for which one or more of the heads reduces to identity are then considered. A theorem is also given that relates to the holomorph of an abelian group of order p™ and type (1, 1, .... , 1). The second section considers the substitutions which are commutative with each substitution of a given transitive group. Jordan's theorem on the number of substitutions that are commutative with each substitution of any regular group is generalized so as to apply to any transitive group. Section III relates to the construction of the imprimitive groups whose sub- stitutions permute the systems of intransitivity of the heads according to the metacyclic group of degree p or to one of its transitive subgroups of degree^. The heads considered are : 1), those whose transitive constituents are the sym- metric or the alternating groups of degree « (« > 2) , and 2), those whose con- stituents are transitive subgroups of degree q having a given index under meta- cyclic groups of the same degree. In section IV the results of section III are made use of to determine the imprimitive groups of degree fifteen. Section I. — On the imprimitive groups whose substitutions permute all their sets of systems of imprimitivity according to primitive groups. 1. Let G denote an imprimitive group that has more than one set of systems of imprimitivity, and let the corresponding heads of G be denoted by H^, H^, etc. Suppose, farther, that the systems that correspond to the head Ei are per- muted by the substitutions of G according to the group P^, where i equals 1, 2, ... . Theorem.— 7/ <^e heads H^, R^, are all different from identity, and if the groups Py, Pz,\. . . . are primitive, then KuHN : On Imprimitive Substitution Groups. 49 (a). The heads can have no substitutions in common besides identity, and hence (b). Each substitution of Hi is commutative with each substitution of H^ (i andj being any two of the subscripts of the H's). (c). Each head contains at least one substitution vjhose degree equals the degree ofG. (d). Any head Hi is formed by establishing a one-to-one isomorphism between its transitive constituents. (a). The systems of in transitivity of any head of an imprimitive group are systems of imprimitivity of this group, and they are permuted by its substitu- tions according to a 6xed transitive group. It follows that the systems of imprimitivity of the groups we are considering must be the systems of intransi- tivity of the heads.* Consider now any two of the heads, H^, H^ say. It results directly from what has been stated that 5i and H^ cannot consist of the same substitutions. Let us assume then that H^ is contained in H^. In this case there must be an a, 1 isomorphism between Pj and P^. The elements of any transitive con- stituent of Hi are composed of the elements in a definite number {m say) of the transitive constituents of H^. Let the two sets of systems of imprimitivity be denoted by «i I «a ,•■••, «m ; bi,bi, . ■ . . , b„; c^, c.^, ■ . ■ ■ , c^; .... and a , b , c ..... those in the first row composing the elements of Pj and those in the second row the elements of Pg. The subgroup h^ of order a in P^ that corresponds to iden- tity in Pg can only permute the a's among each other, the b's among each other, etc. That is, A„ is intransitive, and hence Pj is imprimitive. Assume next that H^ and H^ have a common subgroup Hi^ . This group, jffia, is intransitive, and since it is contained in both Hi and H^, it is invariant in G. Its systems of in transitivity are then systems of imprimitivity. These systems, which are different from those of Hi or H^, are permuted by the sub- stitutions of G according to some transitive group P. As Hi^ is contained in Hi and H2, the preceding argument shows that P is imprimitive. Hence, our proof of (a) is complete. * Miller, American Journal of Mathematics, Vol. 21 (1899), p. 305. 50 KuHN : On Imprimitive Substitution Groups. (b). That each substitution of iT^ is commutative with each substitution of Hj {i andy being any two of the subscripts of the H's) follows at once from the theorem : If every operator of a group Gi transforms the group G^ into itself, and every operator of G^ transforms (rj into itself, then when G^ and G^ have only identity in common every operator of G^ is commutative with every opera- tor of Gi* (c). Let g, hi and Pi be the orders o{ G, Ei and Pi respectively; let, further I, S^, S3, . ■ ■ ■ , Sg be the substitutions of G, and be those of H^ . Form the rectangular array 1 , ^i, . . . ■ , Oftj , from the substitutions of G. None of the rows in this array can contain more than one substitution that belongs to any head different from H^. For if there were two substitutions in any row that belong to H^ (aay), then the inverse of one of them multiplied by the other would give a substitution, different from identity, that belongs to both ^1 and H.^. This, however, cannot be true from what has just been proved. Further, to each row there corresponds one substi- tution of the group Pi. Hence, it follows that any head that differs from H^ is simply isomorphic either to Pi or to some invariant subgroup of Pj . Now an invariant subgroup of a primitive group is transitive, and every transitive group contains substitutions whose degree equals the degree of the group. Hence every head different from -Hj contains substitutions whose degree equals the degree of G . Similarly, by writing the substitutions of G in rectangular array with respect to the head jBg, we see that H^ contains substitutions whose degree equals that of G. (d). We have seen that any head, H^, that is different from H^, is simply isomorphic to Pi or to some invariant subgroup of Pi . Denote by Qi that sub- group of Pi to which Hz is simply isomorphic. We shall prove now that the transitive constituents of R^ are simply isomorphic to Qj. To establish this it * Dyck, Mathematische Anpalen, Vol. 22 (1883), p. 97. Ktjhn: On Imprimitive Substitution Groups. 51 IS sufficient to prove that each substitution in ff^ involves elements from each of its transitive constituents. The subgroup Qi is transitive and so contains a substitution that puts any element into any other element. That is, H^ contains a substitution that puts any system of Hi into any other system. Since each substitution of Sj is commutative with each substitution of ^ , it follows there- fore that each substitution of Hi contains elements from each of its transitive constituents. Similarly, each substitution of H^ contains elements from each of its transitive constituents, and hence each constituent of H^ is simply isomorphic to Qi. 2. Let hi denote the order of the head Hi. Theorem. — If the order of G is equal to h-^h^, and if H^ and H^ are the only heads that differ from identity, then when P^, P^, . . . are primitive groups, G has just two sets of systems of imprimitivity. From the argument used in proving the theorem in paragraph 1, it is clear that Hi and H^ can have no substitutions in common besides identity, and hence that every substitution of H^ is commutative with every substitution of H^. It follows that G is the direct product of H^ and H^. We are to prove that G can have no set of systems of imprimitivity that are interchanged by its substitutions according to a simply isomorphic primitive group. If G has such a set of systems, it must be possible to represent it as a prim- itive group. It follows therefore that Hi and H^ must be simply isomorphic simple groups of composite order and that G, when so represented, is of degree hi-* The subgroup Gi of G that gives rise to its representation in the primi- tive form is formed by establishing a simple isomorphism between Hi and Hz- Let the substitutions of G be denoted by the symbols 1, «2i "3' • • • ■ ' ^hl and let them be written in rectangular array with respect to the substitutions of Gi. If the first hi of the above substitutions form the subgroup Gi, we have the arrangement 1 , S2 , ■ ■ ^2hi-l> "z\hi — l! •••• > ^hi^Zhi-ly -^fti * Burnside, Theory of Groups of Finite Order, p. 190; Miller, Transactions of the American Mathe- matical Society, Vol. 1 (1900), p. 70. 52 Kuhn: On Imprimitive Substitution Groups. Denote the i^^ row of this array by A^ where i = 1, 2, , Aj. The symbols A may be taken for the elements of G when represented in the primitive form. Also to each element Ai there corresponds a certain number of the elements of G. That is, the subgroup G^ of G that gives rise to its representation in the given imprimitive form must be some subgroup of G^. The subgroup G^ must in fact be maximal in Gi ; otherwise G would contain a set of systems of im- primitivity that is permuted by its substitutions according to an imprimitive group. Smce G^ is formed by establishing a simple isomorphism between two subgroups of 5"i and H^, it follows that G'g is contained in a subgroup M^ whose order is the square of its order. It is also contained in a subgroup M^ whose order is h^ times its order and which contains M^. It follows that one of the corresponding sets of systems is permuted by the substitutions of G according to an imprimitive group. Hence G has just two sets of systems of imprimitivity. 3. Theorem. — When G is regular and has just two heads that differ from iden- tity, then if Pi, Pg, . • • • are primitive groups, G is the cyclic group of order pq where p and q are distinct primes. The number of sets of systems that belongs to any regular group is equal to the number of its subgroups, not including identity or the whole group.* Hence the two heads that differ from identity must be generated by substitu- tions of prime order and the order of one must be different from that of the other. 4. Theorem. — If G contains more than two sets of systems of imprimitivity, and if El, H^,, ■ ■ ■ ■ are all different from identity, then when P^, P^, .... are primitive groups, (a). The degree of any substitution besides identity of each head is equal to the degree of G . (b). The heads are simply isomorphic abelian groups ; each is of degree p^'" of order p^ and of type (1, 1, .... , 1) where p is a prime and m is a positive integer. (a). We prove first that in any such group the heads can contain, besides ' Djck, Mathematische Annalen, Vol. 32 (1883), p. 89. KuHN : On Imprimitive Substitution Groups. 53 identity, only substitutions whose degree equals that of (?. As in the theorem of paragraph 1, write the substitutions of G in rectangular array with respect to the substitutions oi H^. We know that any other head is simply isomorphic to Pj or to some invariant subgroup of Pj. Consider the two heads E^ and H^ and let Qz and Q^ denote the respective subgroups of Pj to which these heads are simply isomorphic. Both Q^ and Q^ contain substitutions whose degree equals the degree of Pj, since an invariant subgroup of a primitive group is transitive. Further, neither Q^ nor Q^ can contain a substitution whose degree is less than that of Pi. This may be seen as follows: Each substitution of Q^ is commuta- tive with each substitution of Q^, since the heads of H^ and H^ have this prop- erty. Suppose now that Q^ contains a substitution S whose degree is less than that of Pj. Then the group { Q.^, S\ that is generated by Q2 and Sis transitive since Q^ is transitive. As the substitution S is commutative with each substitu- tion of Q, this transitive group will contain an invariant substitution /S' whose degree is less than the degree of the group. This, however, cannot be true, since an invariant substitution of a transitive group of degree n must be regular and of degree n. It follows, therefore, that the degree of each substitution, besides identity of any head that differs from H^, is the same as the degree of G. By writing the substitutions of G in rectangular array with respect to H^, it fol- lows by a similar argument that H^ also possesses this property. (b). The group generated by any two of the heads, H^ and H^ say, must be transitive. If it were intransitive it would form a new head that contains both Ml and H^i and this cannot be true according to theorem 1 of this section. Fur- ther, this group \Hi, H.^\ must contain the substitutions of all the heads of G. For if it did not contain a substitution S of some other head, then [H^, E^] and S would generate a transitive group whose substitutions all have the same degree as that of the group and which contains a number of substitutions that is greater than this degree. This, however, cannot be true. It follows, therefore, that the heads are simply isomorphic to each other. In the group that is generated by tlie substitutions of H^ and S3 are found the substitutions of H^. Also any substitution of H^ or of H^ is commutative with each substitution of E^, and, therefore, any substitution in the group \E^, jffgj- is commutative with each substitution of E^. It follows that E^, and hence also each of the heads of G, is abelian. Further, each head must be of 54 KuHN : On Imprimitive Substitution Groups. order p^ (where ^ is a prime and wi is a positive integer) and of type (1, 1, , 1). For if the substitutions of the heads were not all of the same prime order, then the subgroup of ] iTi , ^^ [ that is generated by its substitutions of lowest order would form a head that would not satisfy the requirements of the theorem in paragraph 1. Finally, since the group {H^, H^\ is regular, it follows that the degree of G is p^"^. Corollary : If G is regular it must he the non-cyclic group of order p^. When G is regular it must coincide with the group generated by any two of its heads. It is then an abelian group of order jp^"" and of type (1, 1, . . • • , l). The only groups of this type that satisfy the requirements of the above theorem are clearly the groups of order p^. 5. Let P denote an abelian group of order p" and of type (1, 1, . . . . , 1) when represented as a transitive group in the elements With each substitution of P associate that element which replaces a^ in that substitution. The group of isomorphisms of P may then be represented as a transitive group in the p^ — 1 elements a^, a^, .... , a^ ; when so represented, let it be denoted by R. The transitive group (Ji) that is generated by the two groups P and R is simply isomorphic to the holomorph of p. Now to any subgroup of R whose degree is less than p^ — 1 , there corresponds an imprimitive subgroup of h. For such a subgroup of R would transform some of the substitutions of P into themselves, and these would form an invariant intransitive subgroup of the corresponding subgroup ofh. Further, to any transitive subgroup {R^ of R whose degree equals jp" — 1 , there corresponds a primitive subgroup (Aj) of A. For R^ is the subgroup of hi that leaves one of its elements fixed ; since this is transitive, it follows that h^ is primitive It remains to consider those intransitive subgroups (/) of R that are of degree p™ — 1 . We note in the first place that if the subgroup (h!) of h that corresponds to / is primitive, then any substitution of / besides identity must contain elements from each one of its transitive constituents. Suppose that KuHN : On Imprimitive Substitution Groups. 55 some of the substitutions of / that diflFer from identity do not contain any ele- ments from a given one of its transitive constituents. These substitutions form an invariant subgroup (/') of / whose degree is less than p"" — 1 . The substitu- tions of P that correspond to the elements of / that are not found in /', form with identity a subgroup of P. This subgroup is invariant in h', and hence the latter would be imprimitive. Hence, when h' is primitive, /must be formed by establishing a simple isomorphism among its transitive constituents. Every sub- group / of this kind that is contained in B does not give rise, however, to a primitive subgroup of the holomorph. For example, when p equals 2 and m equals 4, R contains a subgroup of order 3 and of degree 15 that gives rise to an imprimitive subgroup of order 48 in the holomorph. The substitutions of P that correspond to the elements in each transitive constituent of the given sub- group of order 3 generate subgroups of order 4. In general, it is evident that the necessary and sufficient condition that h' is primitive is that the elements of each transitive constituent of / contain a set of independent generators of P. If the substitutions that correspond to the elements of any transitive constituent of / generate a subgroup of P of order p° where a is less than m , then this sub- group is invariant in 7i' and the latter is imprimitive. Hence we have this Theorem. — A subgroup (h!) of the holomorph {h) of P, that corresponds to a subgroup (Pj) of the group of isomorphisms (B), is primitive (a). When B^ is a transitive subgroup of degree p^ — 1 , or (b). When Bi is an intransitive subgroup of degree p"^ — 1 that is formed by establishing a simple isomorphism among its transitive constituents and that is such that the elements of each of its transitive constituents contain a set of independent gen- erators of P. Any other subgroup of B gives rise to an imprimitive subgroup of h. Corollary : When the number of elements in each transitive constituent of B is less than m , the subgroup h' is imprimitive. This follows at once from the above theorem, since the number of operators in a set of independent generators of P is m . 6. The theorem just stated is useful in the determination of the primitive groups of degree 16. For it is known that every primitive group of this degree 56 Kuhn: On Imprimitive Substitution Groups. that does not include the alternating group, contains an invariant abelian sub- group of order 16 and of type (1, 1, 1, l).* 7. By means of the two preceding theorems we can investigate the number of distinct imprimitive groups (7 of a given degree that have more than two sets of systems of imprimitivity (the heads differing from identity), and that have all their sets permuted according to primitive groups by the substitutions of G. Any such group is of degree p^™, and the transitive constituents of any of its heads are abelian groups of order p™ and of type (1, 1, ,1). Also each head is formed by establishing a one-to-one isomorphism among its transitive constituents, the number of these being ^'". Denote the transitive constituents of the head H^ by the symbols -o-i, A^, .... Ap„. It follows that Pi is a transitive group in these symbols that contains a regular abelian group {P) of type (1,1, ,1) as an invariant subgroup. That is. Pi must be some subgroup of the holomorph {Ji) of P' that contains P', A being represented as a transitive group in the symbols A^, A^, . . . . , A^,. We consider then those imprimitive groups that contain the head fii and whose sys- tems of imprimitivity that are determined by E^ are permuted according to the primitive subgroups of h that contain P ; the groups G must be found among these. Let the substitutions of H^ be denoted by the symbols 1, «2, /O3 , S . It is assumed that Sj replaces a^ by a^, and that the substitution of the con- stituent A^ that occurs in Sj is found by replacing the element a^. of the substi- tution of J-i that occurs in it by the corresponding element of Ai, where A; = 1, 2, . ■ ■ , p^ and i, j are any two of these numbers. The substitutions that interchange the transitive constituents of H^ in the simplest way according to the substitutions of P, form a second head H^. Let the substitutions of this head be denoted by the symbols 1 , fj , S3 , . . . . , fprn , * Miller, American Journal of Mathematics, Vol. 20 (1898), p. 229. KuHN : On Imprimitive Substitution Groups. 57 tj being that substitution of ff^ that corresponds to the substitution of P' that replaces A^ by Aj. Now, any third head R^ must be formed by establishing some simple isomorphism between the substitutions of H^ and ff^. Without loss of generality, we may assume that the substitutions of H3 are represented by the symbols 1, AJgtj, /O3C3, . . . . , Opmtpm- For let H3 denote the group formed by establishing some other isomorphism between H^ and H2. A certain permutation of the symbols that denote the sub- stitutions of Sj will change H3 into S3, and corresponding to this permutation there is a definite substitution {S') in the group of isomorphisms of A^. Let S[ denote one of the substitutions of the holomorph of Ai that corresponds to S' (the holomorph being represented transitively of degree ^"'), and let S.l denote the same substitution in the elements of the constituent J.^ where i = 2, 3, ... , j)". Then the substitution S[ /S^ Sp„ transforms Hi and H^ into themselves and H3 into H3. That is, a group that contains the heads Hi, H^ and H^ can be transformed into one that contains the heads Hi, H^ and H^. The largest group within which Hi is invariant without having its systems of intransitivity interchanged is the group generated by the holomorphs of each of its transitive constituents Ai, A^, .... , Ap„, these being represented as trans- itive groups of degree ^3™. The order of the resulting group divided by ^9™ gives then the number of sets of substitutions of p" each that permute according to each substitution of Pi. When P] is regular, that is when G is regular, it is clear that m must equal unity. If m were greater than unity, G would contain systems of imprimitivity that are not permuted by its substitutions according to primitive groups. When Pi is not regular, let S' denote any substitution in P that is not con- tained in P' ; and let if denote the substitution that interchanges in the simplest manner the transitive constituents of Hi according to S'. The substitution S' transforms the substitutions of P' according to a certain operator in the group of isomorphisms of P' ; and t' transforms the substitutions of H2 in exactly the same way. Further, t' is commutative with each substitution of Hi. Suppose now that any one of the substitutions which with Hi generate the sets of substitutions that permute according to S' be denoted by 58 Kuhn: On Imprimitive Substitution Groups. where s^ is some substitution in the holomorph of Ai, i being any of the numbers 1, 2, ,^"'. Since the substitution (Aj must transform E^ into itself, it fol- lows that must be commutative with each of the substitutions ti, where i is as above. Hence s^ must be the same substitution in the elements of Ai as s,- is in the ele- ments of J.J, i andy being any two of the numbers 1, 2, , p^. The number of substitutions (B) is then not greater than the number of operators in the group of isomorphisms of A^ . Further, since (A) must transform H^ into itself, it follows that the substitution (B) must transform the substitutions of H^ in exactly the same way as t' transforms the substitutions of B2 . That is, (B) is a fixed substitution. It is clear also that the corresponding substitution (A) thus found has its proper power in the head H^. Hence there is one and only one generating substitution that permutes according to S' that transforms ^1, H^ and H^ into themselves respectively, and that has its proper power in H^ . That is, there is just one imprimitive group of the given kind that is isomorphic to any primitive group P^. The number of primitive groups that contain P' is given by the preceding theorem, so that we have determined now the number of imprimitive groups G of degree ^^". Of the groups G thus found, it is evi- dent that to conjugate subgroups (Bi) of the group of isomorphisms correspond conjugate groups G. We have then the Theorem. — The number 0/ imprimitive groups Go/a given degree that contain more than two sets of systems of imprimitivity (the heads differing from identity) and for which Pi, P2, ■ ■ ■ ■ are primitive groups, is as follows : (a). When G is regular, there is just one such group ; its degree is p^. (b). When G is not regular, the number is equal to the number of distinct prim- itive groups that are contained in the holomorph (h) of the abelian group P' and that contain P'. It is well known that the group of isomorphisms of an abelian group of order p" and of type (1, 1, ... . ,1) is simply isomorphic to the linear homo- geneous group.* It appears then that the study of the imprimitive group G of the above theorem is closely associated with that of the linear homogeneous group. * Moore, Bulletin of the American Mathematical Society, Vol. 3 (1895), p. 34. KuHN : On Imprimitive Substitution Groups. 59 8. We proceed now to investigate the number of sets of systems of imprimi- tivity that belong to the groups just determined. When O is regular, it is the non-cyclic group of order p^ and so contains p -\- I sets of systems — this being the number of subgroups differing from identity that (r contains. Suppose then that (r is a non-regular group. Let B^ denote the sub- group of Pi that leaves any element fixed. We note first that G does not admit the head identity. The subgroup ( G^ of G that leaves any element fixed has one and just one substitution in common with any division of G — the divis- ions being formed with respect to the subgroup that contains the heads H^, ifj , . . . . If now G admits the head identity, then G^ must be contained in a larger subgroup of G which contains no invariant subgroup of G. This is evidently not the case. Any head of G that difiers from E^ and H^ is then formed by establishing some simple isomorphism between these two heads. Denote by H^ any such isomorphisms that differ from H^. Our problem is to determine under what conditions H^ will be transformed into itself by the sub- stitutions of 6r. Corresponding to 5^ there is a definite isomorphism of ^2 to itself, viz., the one formed by replacing each S in E^ by the corresponding t. And this again corresponds to a definite isomorphism of P' to itself. That is, to Ei there is associated in this way a definite substitution (r) of the group of iso- morphisms of P'. Also when E^ is transformed into itself by the substitution of 6r , it is clear that r must be transformed into itself by the substitutions of R^ ; and conversely. It follows, therefore, that the number of heads E^ is equal to the number of substitutions differing from identity in the group of isomorphisms of P' that are commutative to each substitution of i^j. To identity corresponds the head E^. Hence the Theorem. — The number of sets of systems of imprimitivity of the groups G in the preceding theorem is as follows : (a). When G is regular, there are ^ + 1 different sets. (b). When G is non-regular, there are c -|- 2 different sets, where c is the num- ber of substitutions in the group of isomorphisms of P' that are commutative with each substitution of R^ . 9. We consider now the imprimitive groups G that contain more than one set of systems of imprimitivity, each set being permuted by the substitutions of 60 Kuhn: On Imprimitive Substitution Groups. G according to a primitive group, and that have all their heads identity except one. Suppose in the first place that G denotes a regular group of this kind. We know that every regular group contains a set of systems of imprimitivity that is permuted by its substitutions according to any transitive representation of the group. It follows then that G must be a group that can be represented in the imprimitive form only when its degree equals its order, and that can be repre- sented in the primitive form of a lower degree. The only groups that satisfy these requirements are the non-cyclic groups of order pg, where p and q are dis- tinct primes.* lip^q, these groups contain q+l subgroups, not including identity and hence G contains q + 1 sets of systems of imprimitivity. The heads that correspond to q of these are identity and the remaining head is of order p. Hence we have the Theorem. — The only regular groups G that contain more than one set of sys- tems of imprimitivity, each set being permuted by the substitutions of G according to a primitive group, and that have only one head different from identity, are the non- cyclic groups of order pq , where p and q are distinct primes. If p'^q , these groups have q + 1 sets of systems of imprimitivity. 10. Suppose next that the subgroup {Gi) of G that leaves one element fixed is contained in the head (H) that differs from identity. When Gi is of order unity, the groups G are determined by the preceding theorem. Also when G is regular, the subgroup Gi is maximal in 3. We prove now that this is true when G is non-regular. If Gi is not maximal in H, there is a subgroup ffi of H in which (tj is maxi- mal. This subgroup H^ determines a set of systems of imprimitivity of G. Two cases arise according as Hi contains an invariant subgroup of G or does not con- tain such a subgroup. In the former case the set of systems that corresponds to Hi is left unchanged by the substitutions of the invariant subgroup contained in Hi. The group G would contain then two heads that differ from identity ; this is contrary to the assumption made. In the latter case the set of systems that corresponds to Hi is permuted by the substitutions of G? according to a group (P) * Dyck, Mathematische Annalen, Vol. 23 (1888), p. 101. KuHN : On Imprimitive Substitution Groups. 61 that is simply isomorphic to O. Since R^ is not maximal in G, the group P would be imprimitive. It follows, therefore, that Gi is a maximal subgroup ofH. Suppose now that H is contained in a subgroup (E') of G whose order exceeds that of H. The subgroup H' gives rise to a set of systems of imprimi- tivity that are found by uniting the systems that correspond to H into larger systems. This new set of systems is left unchanged by the substitutions of H, and so in this case G contains two heads that differ from identity. It follows accordingly that H is a, maximal subgroup of G. That is, the quotient group GjH'is of prime order j9. The systems of imprimitivity that are left unchanged by H are permuted by the substitutions of G according to a primitive group that is simply isomorphic to the quotient group GjE.. Since these systems are the systems of in transi- tivity of H, it follows that the number of transitive constituents of R is equal to^. Consider now a set of systems that is not left unchanged by any substitu- tion of G besides identity, and let P denote the primitive group according to which this set of systems is permuted by the substitutions of G. The subgroup P' of P that corresponds to the subgroup 5" of (r is transitive. Suppose now that IS is not formed by establishing a simple isomorphism among its transitive constituents. It contains then an invariant subgroup whose degree is less than the degree of H. The subgroup of P' that corresponds to this invariant sub- group of H would be of a lower degree than the degree of P ; also it would be invariant in P'. This, however, cannot be true, since an invariant subgroup of a transitive group is of the same degree as that of the group. Hence H is formed by establishing a simple isomorphism among its transitive constituents. The subgroup of G that gives rise to the primitive representation P just considered, must contain (ri as an invariant subgroup, and its order is equal to pg^, where g^ is the order of Gi. Its order could not be greater than pg^ since then 6?! would not be maximal in H. It follows that the degree of P is equal to the degree of each of the transitive constituents of H, so that to each ele- ment of P there are associated p elements of G. Also since G-^ is maximal in E, it follows that P' is a primitive group. When G is not regular, there is just one subgroup besides E that contains Gi- That is, there is just one set of systems that is permuted by the substitu- 62 KuHN: On Imprimitive Substitution Groups. tions of Gr according to a simply isomorphic group. It follows, therefore, that when G is not regular, it contains just two sets of systems of imprimitivity. These results may be summed up in the Theoeem. — If G denotes an imprimitive group having more than one set of sys- tems of imprimitivity, each set being permuted according to a primitive group by the substitutions of G , and if just one of the heads {H) differs from identity, then when the subgroup ( G^) of G that leaves any element fixed, is contained in H. (a). The head H is maximal in G, the quotient group G/H being of prime order p . Also H is formed by establishing a simple isomorphism among its transi- tive constituents — the number of these being p ■ (b). The subgroup G^ is maximal in H. (c). G has just two sets of systems when it is nut regular. The one whose head is identity contains p elements in each system. 11. It is not diflBcult to construct general classes of groups that come under those defined in the preceding theorem. Suppose, for example, that the transitive constituents of H are alternating groups of degree n (n > 2) and that R is formed by writing after each substitu- tion of the first constituent the same substitution in the other constituent. R has just two transitive constituents. A substitution which with R generates a group G of the kind in question cannot be the ordinary t ; for this would be a commutative substitution and the resulting group could not be represented in primitive form. Let s^ denote any negative substitution of the symmetric group of degree n that contains the first transitive constituent of R; and let s^ denote the same substitution in the elements of the second constituent. Then s-^s^t (A) is a substitution which with R generates a group G of the desired kind. Fur- ther, there is just one group of this kind. For any other substitution that could be used in place of (A) would be of the form where S^^ is some substitution in the elements of the second constituent that is commutative with each substitution of this constituent. Since the constituents are primitive groups there is no such substitution except when n = 3 . In this latter case the group is regular and there is just one. KuHN : On Imprimitive /Substitution Groups. 63 A second class of groups G is obtained in a similar way if we take for the transitive constituents of H semi-met acyclic groups of degree q. The groups thus obtained are of degree 2q, and they are simply isomorphic to the metacy- clic groups of degree q . 1 2. Let, finally, G denote an imprimitive group that contains no head that differs from identity and that has all its sets of systems of impriraitivity per- muted by its substitutions according to primitive groups. Denote by Gi the subgroup that leaves a given element unchanged. The simple group of order 60, when represented as a transitive group of degree 12, illustrates the occurrence of a group of this kind that has just one set of systems. When the same group is represented as a transitive group of degree 20, it has two sets of systems, both of which are permuted according to primitive groups. In these examples the subgroup Gi is maximal in larger sub- groups of G. In general, it is evident that the subgroup G^ must be maximal in any larger subgroup ( G^ of G in which it is found, and, further, that the subgroup (3*2 cannot contain any invariant subgroup of G besides identity. It is clear that G cannot be regular. Also its order cannot be the power of a prime. The num- ber of sets of systems that belong to any such group is equal to the number of subgroups Gz that it contains. 13. Theorem. — Every imprimitive group G that admits only the heads identity is insolvable. Since G admits only identity as a head, it must be possible to repre- sent it as a primitive group (P) . If now G is solvable, so also is P- Suppose that G is solvable. Then P must be of degree p", where ^ is a prime, and it contains as a minimal invariant subgroup an abelian group (Pj) of order jf and type (1, 1, . . - . , 1) .* To the subgroup Pj of P there corresponds an inva- riant subgroup of G ; and since Pj is regular and of degree p°-, it follows that this subgroup is also intransitive. Hence G in this case contains a head that diifers from identity, and this contradicts our hypothesis. • Galois, Oeuvres Mathematiques, p. 37. Cf. also Jordan, Traite des Substitutions, p. 398. 64 Kuhn: On Imprimitive Substitution Groups. Section TI.— 7%e substitutions which are commutative with each substitution of any transitive group. 1. Theorem.— ie< G denote any transitive group of degree n and order g. The number of substitutions in the elements of G that are commutative icith each of its substitutions is equal to the order of the quotient group H/G^, where G, is a subgroup of G that leaves any element fixed, and His the largest subgroup of G that contains Gi self-conjugately. It is well known that this theorem is true when G is regular* To prove that it is true for any transitive group G, let the substitutions of the group be denoted by 1, Oj, A3, • ■ - • , Wj,) and the elements of these substitutions by ftj , (Zg , . . • • , a^ • Suppose that G^ is that subgroup of G which leaves a^ fixed, and let its substitu- tions be denoted by 1, A3, O3 , • . • • , Og^ , where g^ is the order of Gi. The substitutions of G may be arranged in the following rectangular array : 1 , ^i , • ■ • ■ y ^g, > «1. "gi + 1 ' ^Z^gi+l ' • > ^gi^g.+ n-li «» In this array the substitutions of the i^^ row replace a^ by a^ where i is any of the numbers 1,2,. . . , n. Also if the i*'^ row is denoted by a^, then any substi- tution ySj of G is given by the permutation on the symbols associated with the rows which arises when all the substitutions of the array are multiplied by Sr^.f If the substitution S^ is commutative with each substitution of G, then evidently the same permutation of the rows will take place whether pre-multiplication or post-multiplication is made use of. * Jordan, Journal de I'Ecole Polytechnique, Vol. 23 (1861), p. 153 ; cf. Traite des Substitutions, p. 60. t Miller, Bulletin of the American Mathematical Society, 3d series, Vol. 8 (1896), p. 214. KuHN : On Imprimitive Substitution Groiq^?. 65 We prove now that any substitution G, which is not found in G and which is commutative to each of the substitutions of G, is given by the permutation on the symbols associated with the rows that arises when some substitution of G is multiplied by all its substitutions , i. e., by pre-multiplication of some substitu- tion of G. Since G is not found in G, it will generate with G a larger transi- tive group & in the same elements. The subgroup Gi of G' that leaves a^ fixed will contain G^. Let SfG denote a substitution which with G^ will generate &[, Sf being some substitution of G. As above, the substitutions of G' may be , SfC ,...., Ui , ) cf, + 1 ' StGSg^^i , .... ,a2, 1 arranged in the following array : 1 , S2 , . . . . .'^. '-'ffi + 1 > "2 '^ffi + 1 , • • • • ' ^g, ^0 '-^Sr, + n-l> W2*J!7i + n-l' •••• • '^g. ^i ^ — ] I *->( ^ »Jgi -I- n — 1 > "n The substitution G is contained in this array, and, as noted above, if the pre-multiplication of O upon the substitutions of G is performed, the resulting permutation on the elements denoting the rows is identical with C. If pre-mul- tiplication be used with reference to any substitution in the first row of the array, then each row goes into itself. This follows at once from the fact that tthe substitutions in the i*"^ row replace a^ by a,-, where i equals 1, 2, . . , n. Suppose now that the substitution S^"^ C~^ G is multiplied by each substi- tution of G'. From what has just been said, it follows that the resulting per- mutation on the letters associated with the rows is identical with C. The sub- stitution Sf^ G~^ G is identical with Sf^, and hence we have proved our statement. We prove now that the number of substitutions of G which by pre-multipli- cation give rise to distinct permutations on the elements associated with the rows of the array I, is equal to the order of the quotient group H/ G^, where H is the largest group that contains G^ self-conjugately. Let SiGi and G-^Si denote the result of pre-multiplication and post-multiplication of G^ with Si respec- tively. Then if S^ by pre-multiplication gives rise to a permutation G of the rows of I, we must have StGi=. GiSg^+1,, StGiSr^= GiSg^+hSr^^^ GiSr, {Sr'= Sg^+^sr^). 9 66 KiTHN: On Imprimitive Substitution Groups . The right-hand member of this identity consists of the substitutions found in some row. And since the left-hand member contains the identical substitution, it follows that St must transform G^ into itself. Conversely, if St transforms G^ into itself, it gives rise to a permutation of the rows of I. It has been seen above that, if St is a substitution of G^ , the corresponding substitution O is the identical substitution. When St is any substitution of H that is not found in Gj, the corresponding substitution C will be different from identity. Also all the substitutions of Hthat are found in the same row with Sf will give rise to the same substitutions G, since these substitutions are found by multiplying those of Gi by Sf . Finally, the substitutions G that correspond to substitutions St that are found in different rows of JS are distinct. For each replaces a^ by the element associated with the row in which it is found. The number of substitutions C is then equal to h, the order of the quotient group R/ G^. It is clear also that the substitutions found in this way are all commutative with each substitu- tion of G. Since His a. subgroup of G, it follows that the number of rows contained in jff is a divisor of the number contained in G. Hence Corollary I. — The numher of substitutions that are commutative with each sub- stitution of G is equal to some divisor of the degree of G . When (r is a primitive group, the subgroup 6?! that leaves one element fixed, is maximal, and in this case the order {h) of Hj G^ is equal either to unity or to gr. Hence we have Corollary II. — Identity is the only substitution that is commutative with each substitution of a primitive group of composite order. 2. If a denotes the number of letters left fixed by G-^, then nja is the num- ber of subgroups in the conjugate set to which G^ belongs. The number of sub- stitutions, X, that transform G^ into itself is the same as the number that trans- forms it into any one of its conjugates. X . n/a = g =ng^ , or a; =: ag^ , or xlgi = h = a. KuHN : On Imprimitive /Substitution Groups. 67 The above theorem may therefore be stated in this form : The number of substitutions in the elements of any transitive group G that are commutative with each of its substitutions is equal to the number of elements that are left unchanged by the subgroup G^ that leaves any element of G unchanged. When the order of G is the power of a prime, then the order of the quotient group S/ Gi is also the power of the same prime. Hence we have Corollary I.* — If the order of G is the power of a prime, then the number of elements left unchanged by Gi is a power > 1 0/ the same prime. 3. When 6^ is a regular group, the substitutions that are commutative with each of its substitutions, form a group that is simply isomorphic to G'. So in this case the order of each substitution that is commutative to all the substitu- tions of G is equal to the order of some substitution of G . When G is not reg- ular, this is still true. For, from the method of forming any substitution G that is commutative to all the substitutions of G, it follows that to C there corre- sponds a definite substitution in the associate of G whose order equals that of G. We have then the Theorem. — The order of any substitution that is commutative with each substi- tution of a transitive group is equal to the order of some substitution of that group. 4. Let the associate of a group G whose order is g be denoted by G'. If G contains no invariant substitution, then it contains no substitutions in common with G. In this case G and G' generate a group \G, G'\ whose order equals g'' and whose degree equals g. The subgroup G^ that leaves any element fixed in \G, G'\, is formed by establishing some simple isomorphism between G and &. For, suppose the substitutions of { G', &\ are written in rectangular array in such a way that the substitutions of G form the first row. The subgroup Gi cannot have more than one substitution in common with any row of this array. If it had two, then one times the inverse of the other would belong to G, and this could not be true since this product would be a substitution differing from identity that is found in G. The group G-^ contains, therefore, just one substi- tution from each row of our array. The substitutions in any row are found by multiplying the g substitutions of G by some substitution of G' . Further, no * Miller, American Journal of Mathematics, Vol. 23 (1900), p. 173. 68 Kuhn: On Imjprimitive Substitution Groups. two substitutions of G^ can involve the same substitution of G ; for then one times the inverse of the other would be of degree g. It follows directly that G^ consists of a simple isomorphism between G and G'. 5. By means of the statement just proved in reference to &»;••••; G^, the sym- * Miller, Quarterly Journal of Mathematics, Vol. 38 (1895), p. 193 ; American Journal of Mathemat- ics, Vol. 21 (1899), p. 295. Kuhn: On Imprimitive Substitution Groups. 69 metric group in the elements m^, m^, m„. Alao let P denote the meta- cyclic group of degree jp in the letters A, B , O, . . . . , ilf and P;^ that invariant subgroup of P whose index under P is i^. When i^ is dififerent from p — 1, the group P^^ may be generated by a substitution {p^) of order p and a substitu- tion (i^a) of order ^-; — ; when i^ equals p — 1 , the group P,^ is generated by a single substitution of order ^. The generator p^^ may be taken as the substitu- tion ABG . . . ■ M, and it may be assumed that the generator p^ does not contain the letter A. When p equals 2, P will denote the group {AB). The substitu- tions that permute the systems in the simplest way according to pi and p^ will be denoted by ti and t^ respectively. That is, ^1 = ai&iCj .... wij . a^b^i .... m^ a„&„c» • ■ • • »*«. and tg is a substitution of order ^T that does not involve the elements h (Xi, 0,2, .... , Cl„. 2. Theorem. — The number of imprimitive groups of degree np that contain the head H={G', a", .... , G^\^08* and whose substitutions permute the systems of intransitivity of H according to P.-^ is as follows : (a). When p = 2, there are two such groups. (b). When p > 2 , there is one group if -?-— ^ — is odd and two groups if -t-—. — is even. The largest group within which the given head is invariant without having its systems of intransitivity interchanged is { G', G", . . . ., G'\. Hence there are just two sets of substitutions that transform according to any substitution of Pj^ . Those that permute according to p^ may be obtained by multiplying the substi- tutions in the head by ti and a^a^ . t^ . * The notation used is that given by Cayley, Quarterly Journal of Mathematics, Vol. 25 (1890-1), p. 71. 70 KuHN: On Imprimitive Substitution Groups. Both of these transform H into itself. The p^^ power of f^ is identity, while that of a,az. ti is a:^az.bj\ m^m^. This latter is found in the head only when p = 2. Hence there is just one imprimitive group that contains the given head and that corresponds to Pp_i when i? > 2. When p = 2, there are two such groups ; these are distinct, since one contains negative substitutions while the other does not. When »2 is less than p — 1, the substitutions that permute according to p^ may be obtained by multiplying the substitutions of H by t^ and aia^ . t^ . Since there is just one group that corresponds to Pp_i, each group that corre- sponds to Pjj must contain this, and hence we may assume that ti is found in each such group. The substitutions t^ and a-^a^ . t^ both transform the head and also the group that corresponds to Pp_i into themselves. The ^ T~ th power of ^2 is identity, and hence there is always one group that corresponds to Pj^. The ^-—. — th power of a-^a^ . t^ is identity or a^a^ according as ^T" is even or odd. Hence a^a^ . t^ may be used only when ^—. — is even. The corresponding group h is distinct from the one obtained when ^2 is taken, as the one contains negative substitutions while the other does not. Hence, when ^~ is even there are h two groups that correspond to P^, and when ^-^^ is odd there is just one such h group. 3. Theorem.— TTAenp > 2, there is just one imprimitive group of degree np that contains the head H= Q' pos G" pos G^ pos + G' neg G" neg G'' neg and whose substitutions interchange the systems of intransitivity of H according to P,„ . When p= 2, there are two such groups. The largest group within which H is invariant without having its systems interchanged is \G', G", .... ,G''\, and hence there are 2"-! sets of substitu- tions that transform according to a given substitution of P^ , The 2^-^ sets KuHN : On Imprimitive Substitution Groups. 71 that transform according to pi, may be found from H and the substitutions obtained by multiplying each substitution in the group K)W • ■■{s,-,) (A) by ti, where Sj = a^a^, s^ = bj)^, etc. Bach of these transforms 5" into itself and each has its ^j*** power in R. The resulting groups, however, are all conjugate when p is greater than 2. For S3S5 SptyS^S^ S, = S2S3 Spt^. That is, the group {H, t^} can be transformed into the group \ff, Siti\. Simi- larly, it can be transformed into the groups { H, Sit^}, where i = 2, 3, , p — 1 and, therefore, into all the others. Hence when jp > 2, there is just one group that corresponds to Pp_i. It may be taken as \II,ti\, ^iid we may assume that it is found in each group that corresponds to Pi^{i2<.p — i)- When^ = 2, the head contains only positive substitutions. In this case the groups that corre- spond to P are distinct, since the one contains negative substitutions while the other does not. By multiplying each substitution in the group (A) by t^ , we obtain a set of substitutions which with H generate the sets that transform according to p^ . Bach of these transforms the head into itself, but 4 is the only one that trans- forms 1^, ti\ into itself. For if we transform t^ by the inverse of any other one, SiSj .... t-i, say, we have S^Sj .... fgtifg *i*; ■ • ■ • — — SiSj .... SjiSji . • • f 1 , where t^t^t^ ^ = tl , and i', f are certain ones of the subscripts 1, 2, . . . . , p. There is an even number of s's in'the part of this substitution that precedes ij. further this part cannot be identity. Hence this substitution is not found in \ff, t^l, and, therefore, there is just one group that corresponds to Pj^ when p>2. 4. Theorem. — When^-—. — is even, there are two imprimitive groups 0/ degree h np that contain the head E= G' pos G^' pos G^ pos and whose substitutions interchange the systems of intransitivity of H according to Pi, • Where ^—. — is odd, there is just one such group. 72 KuHN: On Imprimitive Substitution Groups. The largest group within which H is invariant without having its systems interchanged is { G\ G", , G''}. There are then 2^ sets of substitutions that transform according to a given substitution of Pj,. We know that there is just one distinct group that contains this head and corresponds to a cyclic substitu- tion of order p. Hence it may be assumed that each group that contains H and that corresponds to Pi^ contains the substitution t^. The substitutions that permute according to p^ may be found by multiply- ing H by the substitutions obtained by affixing <2 to each substitution in the group where $i has the same meaning as in the preceding theorem. Of these genera- tors 4 and SjSg .... s^tz are the only ones that transform {H, t^} into itself. The former has its - — -. — th power in H and generates with \n, B and n= 3 or p^2 and n^3, there are two groups if ^~^ h is even and there is one group if^^^^ — is odd. Kuhn: On Imprimitive Substitution Groups. 73 In this case H is formed by writing after each substitution of & pos the same substitutions in the other sets of elements. (a). When p=2 and n = 3, there are evidently two distinct groups. When p = 2 and n > 3, the only substitutions that permute according to p^ and that also transform H into itself, may be obtained by multiplying H by the substitu- tions ^i and Sis^^i. Both of these have their squares in the head and they gen- erate with H groups that are clearly distinct. (b). When^ = 3 and « = 3, the groups that are isomorphic to Pp_i are of order and degree 9. It follows that there are two such groups, since there are two distinct abstract groups of order p^. They may be written \E, ti\ and \H, aia^a^ti}. The substitutions that permute according to p^ and that transform both H and \R, ti\ into themselves may be obtained from the head by means of the substitutions SiS^t-i, SiS^SiS-^S^t^, where Si^ajUias, iSz = bib2b3, etc. Each of these has its square in the head except the last two in the second column. Those in the first column clearly generate with \H, i^} conjugate groups. The groups {-5", ti, t^} and \H, ti, ^iVs^a} ^'^e distinct since one contains only positive substitutions while the other contains negative substitutions. The group {B, ayo-za^t-^] is a cyclic group of order 9. If a group that corre- sponds to P contains it self-conjugate ly, then that group is of order 18. There is just one such non-abelian group of this order,* and it can be represented in only one way as an imprimitive group of degree 9. Hence there is just one group with the head H that contains \H, a^a^Jii] self-conjugately and that cor- responds to P. (c). When ^ >• 3 and n =: 3 , the groups that are isomorphic to Pp_i are of order and degree p.Z. Since j) >• 3 and the subgroup of order 3 is self-conju- gate, it follows that there is just one group that contains 5" and that corresponds to Pp_i. It may be written \H, ^j}. * Cole and Glover, American Journal of Mathematics, Vol. 15 (1893), p. 206. 10 74 Kuhn: On Imprimiiive Substitution Groups. The general substitution that permutes according to pz may be taken of the form Si'S^ . . . . S;^ {s,s, . . . . s^iH,, (A) where a^, a^, , a^ = 0, 1, 2 and /? = 0, 1. If this substitution transforms \S,ti\ into itself, then St' Si' S^ must also do so. Now Si^Si' .... s^\, sr'-'Sr^^ .... s^"= s^^-'-^sr-^' ■■■■ s^-'^t^. If this be in \H,ti\, it must be of the form {SiSi ■ . . ■ SpY ti . That is, we have the relations tti — ai = — (i — l)a (mods), where i=l, 2, . . . . , p and a = 0, 1, 2. Putting i = p, we see that ap=0 (mod 3). Hence zero is the only permissible value of a and of the substitutions (A), we need consider only tcj and Sjfij .... Spt^ , The first of these has its ^T th power in the head and hence generates h with \II, ti] Si group that corresponds to P^^. The second has its ^. — th power h in the head only when -^. — is an even number. h Therefore, when ^-—. — is even, there are two groups that contain the given h head and that correspond to P,-^ . They are distinct, since one contains only pos- itive substitutions while the other contains only negative ones. When ^ > 2 and « > 3 , the substitutions that permute like pi and that transform S'into itself, may be found by multiplying 5^ by the substitutions ti and SiSg ■ . . • Spt^ . Of these only the first has itsp*'^ power in the head, and hence there is just one group that contains the given head and is isomorphic to Pf-i. The remain- Kdhn : On Imprimitive Substitution Groups. 75 ing part of the proof for this case is the same as the latter part of the preceding one. 6. The group \ G', G", , G"} may also be used as a head. It con- tains all the substitutions which transform it into itself without interchanging its systems of intransitivity. Hence it is contained as a head in just one imprimitive group whose substitutions permute its systems of intransitivity according to a given transitive group. The same remark applies to the head {G',G",....,G^\,,, ,.* 7. The remaining theorems in this section apply to imprimitive groups of degree pq where p and q are prime numbers which may be the same or different primes. Let Gi denote the metacyclic group in the elements a^, a^, . ■ . ■ , a^ ; G^, the metacyclic group in the elements bi, b^, ,\; ; 6r,, the same group in the elements mi, m^, • ■ ■ ■ , m^; and let Gi^i^ denote that invariant subgroup of Gi whose index under Gi is tj. The symbols P, P^^, p^, p^, t^ and t^ will be used as they were in the theorems just given. We shall assume further that the group Gi is generated by the substitution /Si = aia^s a^ of order q, and a substitution Si of order q — 1 in the q — 1 elements aj, aj, a^, . . . . , a,. The symbols Si and s,- will denote the same substitutions in the elements of the group Gf, where i = 2, 3 , , p, q.s Si and «i do in the elements of G^ . 8. Theorem. — The number of imprimitive groups of degree pq that contain the head and whose substitutions interchange the systems of intransitivity of H according to Pi^, is equal to the number of solutions of the congruences ^(P-^) = hii (mod^-1), where ^ is restricted to the values 0, 1, 2, .... ,i — 1 and h is any integer. * Miller, Quarterly Journal of MathematicB, Vol. 38 (1895), p. 195. 76 KuHN : On Imprimitive Substitution Groups. The largest group within which E is invariant without having its systems interchanged is {G^, G^, . . . ■ , Gj,\. There are accordingly i? sets of substitu- tions that transform according to each substitution of Pj^. Since the given head is the direct product of p transitive groups written in different sets of elements, we know that there is just one distinct group that contains this head and that corresponds to Pp _ i . This may be taken as {S,ti\ and we may assume that each group to be found contains this as a self-conjugate subgroup. As sf is the lowest power of s^ besides s" that occurs in G^<, -fe + »1 s|«- fe + "= __ gh-?,+-Pf^— g< gai _ gal, f^ _ when 13, = 13, + (a^ + a^+ .... + a^^,) — {a[ + a', + .... + a^i) where i=2,S , . . . ,p. Also sl'sl'- s/" transforms the head into itself. Hence there cannot be more than ^~ distinct groups that correspond to P«_i— one to each of the above congruences, Ktjhn : On Tmprimitive Substitution Groups. 83 The substitutions in the following table give a set of g' — 1 generators— one corresponding to each congruence — which, with the given head, generate one set of*—; — groups isomorphic to Pp-i : (SlSj .... Sp)% , si (SjSg .... Spf ti, . . . . , Si^ - 1) •■i (si Sg . . . . S,)*' ti , ■ S,f%, S{^ {S,S, .... S^f^ t„ .... , Si^-"H (s,S, .... Sj'^ t, , (SiS. 3 the table being continued until ^. — substitutions are contained in it. *i In the first place it is evident that the groups that correspond to the substi- tutions in the i^^ column where i= 1, 2, . . . . , p, are identical. Hence we need consider only those groups that correspond to the substitutions in the first row. In the group ] H, s'^ ti ] are found the substitutions {SiSpSp_i .... Sp _a, + 3) ' ti, where x= 2, S, . . . . , p — 1. If j?2 be of order p — 1 , then a certain power of the substitution ^3 that transforms according to p^ will transform t^ into t^ and the group \H,s^ti\ into a group that is conjugate to {H,sf'ti\. Hence the groups \R, s{'^ti\, where r=l, 2, .... , p — 1 are conjugate. We consider then the two groups \H, ti\ and \H, sl'- t^}. The latter contains in the division in which Sj' ^1 occurs the substitutions Sl'S^,' .... S^^ Sl'^ + ^s^s ■ ■ ■ ■ s.^H,, where a( , a^ a^ = 0, 1, , or 5 — 1 and A is any integer. If the y power of any of these be identity, then {hp+l)ii = (modg — 1), or {hp+l)ii = k{q—l), or JcM—hp =1, (2) if M=(q l)/h' When ilf is not a multiple of p, this equation has a solution. In this case there is a substitution of order p in the division in question. The corresponding exponents of the s's then satisfy the first congruence of (1), and the group is conjugate to \E, t^}. When, however, ^T = mp, where m is an 84 KuHN : On Imprimitive Substitution Groups. integer, then (2) becomes [hm — h) p:= \ . This has no solution. Hence when {q — l)/ii is not a multiple oi p, there is just one group containing the given head that corresponds to Pp_i ; but when {q— l)/*i is a multiple of p, there are two such groups. The general form of the generating substitutions that transform according to p)^ is sr«r •••■ «?<2. If any of these transform \H,t^\ into itself, then sl'Si' . ■ ■ ■ s^ must do so. That is, s^'s^ . .. . s^ ^iSf"' s^"" . . . . s^"' = Si'-°^S2^-"^ sf ~''Hi must be of the form where a is some multiple of ii. That is, we have the relations ai = ai — (i — 1) (mod 5- — 1). Putting i ^ jp , it is seen that ap=0 (mod g' — 1), p Hence, as the values of a to be considered are multiples of i-^ that are less than q — 1, it follows that q — 1 must be a multiple of pi^ unless a = 0. In the latter case, aj = aj =: .... =■ ap. That is, when [q — l)/*i is not a multiple of p, the substitutions that permute according to p which we have to consider may be obtained by multiplying the head by the substitutions (SiSg SpY t^ , where /3 = 0, 1, 2, . . ij — 1. When, however, {q — l)/ii = mp, we must mul- tiply the head by where a == 0, mi^ , 2mi^, . . . . , [p — 1) wjij . These may be written in the form oCp-iji-cp— 2)« o^^o o o ^•l / *2 °S • • • • 6p ^^6j Sg • • • • Sp) Ij. KuHN : On Imprimitive Substitution Groups. 85 If a( is a value of aj for which the [p ~ \)li^^ power of this substitution is in the head, then there cannot be more than p groups that correspond to this value of aj. We prove that there is just one— i. e., that the groups that corre- spond to the substitutions 6jj O3 . . . . Sp ^^6182 ■ ■ ■ ■ Sp) Ig are conjugate. For, transform the substitution (sjS^ • - • ■ Sp)°'<2 by the inverse oisf-^^'^sf-^^'^ .... si. This latter transforms \H,t^\ into itself. Suppose that f^^ transforms S^ into S^. Then the transform in question is of the form Sg " . . . . [SiScj .... Sp)"' t^ . When a takes the values 0, mi^ , . ■ ■ , {p — 1) mi^ , we getp diflferent sub- stitutions in this way. For, let imi^ and jmii be any two distinct values of a . If these gave rise to the same substitution, then we would have [x — 2) imii ={x — 2)jmii (mod q — 1), or {x — 2)(i — j^mii = k{q — 1) =: kmpii, or (a; — 2)(i —J) = hp. 03 — 2 = (mod ^) , X =2 (mod p) . This, however, is not true. Hence, in any case, the substitutions that transform according to P2 may be found by multiplying Hhj the substitutions (SjSa .... Sp)i t^ , where (3=^0, 1, ■ • • , *i — 1- As above (§8), we prove that to each value of |8 that satisfies the congruences /? (^^^) = hi, {modq—1), there is a distinct group that contains \H, t,\ as an invariant subgroup. It remains to consider what (if any) groups contain {ff, Sl't,} as an inva- riant subgroup which corresponds to F„. The general substitution to be con- sidered is, as before, Sl^ S^ .... S^ Cg . Transforming s'^ t^ by this, we get — s^ - .1 + '1 ^«a - «. s.°4 - «3 s.«i - « t-{, (Ai) 86 KuHN : On Imprimitive Substitution Groups. where the subscripts 4, ig, .... i^ denote the numbers 2, 3, . . . . , p in some order and where / is defined by t^H:^t^ = tl'. If this be a substitution in {H, s\' ti\,it must occur in the division in which (sj tiY or (s^Sp .... Sp_^r+2)''^];' is found. In this division we have the substitutions {s,s,.... s^Y'is, s,....Sp _y + ,y^ tj'. (Ag) If (Aj) be identical with any of these, then the j-^^^ power of each must give rise to the same substitution. Now the p*^ power of (Aj) is (s-^s^ .... s^)'', while that of (Ag) is (s^s^ .... 5^)^*^ + '>'''''. Hence we must have the relation {hp + y') ii = % (mod ^ — 1) , or {hp + y') ii = (1 + kmp) ij, -j since ^-^ — = mp L or y' = 1 + {km, — h)p. This, however, cannot be true, since y' is not congruent to unity modulus jp. Hence there is no group that contains ] H, s/' t^^ \ as an invariant subgroup and that corresponds to P^^, i^ being different from p — 1 . 11. Let S"ij denote that invariant subgroup of {S,S^'){S,St') .... {S,_,S-')(s,s, ....,) whose index under G is iy , p being an odd prime. Theorem. — The number of imprimitive groups of degree pq that contain the head 5-, and whose substitutions interchange the systems of intransitivity of Hi according to Pi^ is as follows : (a) When ii'=p — 1 and p=^q there are two groups if ii = q — 1 and one group ifii — 1- Hence the number of dis- tinct groups that contain |5p_i, ^i} as an invariant subgroup is equal to the number of solutions of the congruence (1). If (siSg SpfH^ transforms \Hj,_-^Si t^\ into itself, so also will Si' {s^Si s^fH^. We need consider then only those substitutions in the first row whose exponents satisfy (1). Now if ^^ be such a value of the exponent, then tj^ {s^s, .... s,)-'' S,t, {s,s, . . . s,y^ t, = ST'' S,st' tf = Sr tl\ where tj^t-f^^^tf and sY^'S^sl = Sf. This will evidently be a substitution in {Sp_i, Si ti\ only when a' = /i- Hence, there is just one substitution in the first row that transforms the given group into itself — the corresponding value of /? being i^. All the substitutions in the corresponding column give rise to conju- gate groups, as may be seen by transforming \R, S^ti, (sjs^ .... SpYH^] by S^, where a = 1, 2, .... ,p — 1. Hence, there is just one group that contains \B, Siti\ and that corresponds to Pj^. (2). When ii<,q—l. In this case, the largest group within which the given head is invariant without having its systems interchanged is {SA-'){s,Sr') .... {s,_,s-'){sis, .... s^). There are then i^ gets of substitutions that transform according to any substitu- tion in Pij. Those that transform according to Pp_i may be obtained by multi- plying the head by the substitutions (s^Sg .... Sj,y ti , where a =0, 1, . . . . , ii — 1. If the p*'^ power of any of these be in the head, KuHN : On Imprimitive Suhstitution Groups. 91 then the corresponding a must satisfy the relation ap = Ml (mod q — 1) , where h is as usual. That is, ap:={h + 7eM)ii J£^=if|. When ii is not a multiple of p, zero is the only value of a that can be used; when ii = mp (where m is an integer), then a = 0, m, 2m, ..... (^— l)m. Hence, as before, when p is prime to ij, there is just one group that corresponds to Pp_i, and when ii = mp, there are two such groups. The substitutions that transform according to p^ result when the head is multiplied by the substitutions [S1S2 . . . ■ 8p) t^, where ^ = 0, 1, . . . . , ii — 1 . Bach of these transforms both the head and \Si^, ti\ into themselves. The number of groups that contain {Ili^, ti\ self-con- jugately and that correspond to P;, is then equal to the number of solutions of the congruences (3 r^^") = hii (mod q—1), where h is as usual. By the reasoning used above, it follows that there is no group isomorphic to Pi^ that contains \Hi^, (siSg • • • • s,)"* ^1 } as an invariant subgroup, where h

(Aj) where ^ = 0,1,. . . , q—2. Each of these transforms the head and also \H,li\ into themselves. The ^T th power of (Aj) will be in the head if /3 satisfies h the relation ^(^^) = (mod^-1). (2) To each value of (3 that satisfies this congruence there corresponds a distinct group that corresponds to Pi^{i3'^+^ sp' + ^ sp' + ^'ti. The p*^ power of this will be identity if the y's satisfy the congruence ^{71 + 72+ ■■■■ +rp) + P+^ = ^ (mod^— 1) i.e, y^ + y^.... ^y^z=z-R±l (^mod SL^-l) . KuHN: On Imprimitive Substitution Groups. 95 This evidently has a solution except where p^=2. Hence, where i> > 2, the group \H, s^t^} contains a substitution of order p in the division in which sj^ occurs. In this case, the groups \H,ti\ and \H,s^ty\ are clearly conjugate. When ^ = 2, the group ] 5", Sj jfj [ contains negative substitutions, while \H, t^\ does not. Hence, when p is even, there are two groups with the given head that correspond to Pp_x^ and when p is odd there is one such group. The sets of substitutions that transform according to p.^ may be found by multiplying the head by the substitutions s^'sl^ .... s;^ t„ (B) where aj, aj, . . . . , ttp = or 1 . These all transform the head into itself. Now «r'sr°'C=- . . . s-'U^s^si' ....s;^t, = sp str '%'-"' ..■■ s^z "^-^ K" tl (C) where i^, is, . ■ ■ .i^ = 2, 3, .. . ■ , p in some order and tf^ t^ t^=^tl. Since a2, . . . . , ttjp = or 1, it follows that the exponents of the s's that precede tl are either , 1 , or — 1 . Further if (C) is found in \H, ti\ when the exponent of one of the s's is zero they must all be zero. It follows that a2 = a3= .... =ap=:0 is the only set of values which give a substitution (B) that transforms {H, ti\ into itself. Hence there is just one group that contains the given head and corresponds to P^^ when \ is less than p — 1. 14. Theorem. — When ^ !> 2 and *-—. — is even there are two imprimitive groups that contain the head H= {Gi, Gz, , Gp} pos, and whose substitutions interchange the systems of intransitimity of H according to Pi^ ] when ^-—. — is odd there is just one such group. When p = 2 there are two groups that contain the given head. 15. The heads considered in paragraphs 8-14 occur for all values of q and hence the theorems proved enable us to determine certain imprimitive groups of every degree of the form p q. In general, other intransitive groups can be formed from the p transitive groups Gi^i^, G-^^i^, . . . . , Gp^t^ which may be used as heads of imprimitive groups whose systems of imprimitivity are permuted according to P^^. For each such head a like theorem may be proved. 96 KuHN : On Imprimitive Substitution Groups. Section IV. List of the imprimitive groups of degree fifteen. The theorems proved in the preceding section enable us to find at once most of the imprimitive groups of the degrees four, six, nine, ten, and fourteen. We shall now make use of them in determining the imprimitive groups of degree fifteen. Fifteen letters can be divided in two ways into systems containing an equal number of letters, viz., into three systems of five letters each, or into five systems of three letters each. We consider first the groups that contain three systems of imprimitivity. The substitutions of these groups permute the systems accord- ing to either the symmetric group or the alternating group of degree three. It follows that any intransitive group of degree fifteen that can be used as the head of such a group must permit a cyclic interchange of its systems of intransitivity. It is not difficult to construct all the intransitive groups of degree fifteen having three systems of intransitivity that have this property. It is found that there are twenty-one such groups which can be used as heads. They are as follows: Order. 1728000 (abode) all (fghij) all (Jdmno) all 864000 {(abode) all {fghij) all (Hmno) all} pos 432000 (abode) pos (fghij) pos (Hmno) pos -f (abcde) neg (fghij) nag (Mmno) neg 216000 (abode) pos (fghij) pos (Mmno) pos 8000 (abcde)^^ (fg^ij)io (Mmno)^ 4000 \(abcde)2a (fghij)^ (hlmno)^\ pos 2000 [](a5c(fe)jo (fghij)^^] pos, (Mmno)i^'\ dim 2000 ] (abcde)^ (fghij)io , (Mmno%o \ loo, b 1000 (abcde)io (fghij)i„ (hlmno)^^ 600 { (abode)^^ (fghij)2o (Tdmno)^^ [ 5, b, g 500 \(abcde)^^ (fghij)^Q, (Mmno)if,\ dim 250 \ (a6ccZe)io {/9^v)io (^^mno)^^}^, 5, 5 125 (abode)^ (fghij)^ (hlmno)^ 1 20 (abode .fghij . klmno)i2Q 100 l\ (abcde)^^ (fghij)io U, b. (Mmno)^ak 1 6 (abode . fghij . Mmno) jo 50 [\(abcde)io {fghij)w\ dim, (Mmno)^^^^ KuHN : On Imprimitive Substitution Groups. 97 25 ] {abcde\ {fghij \ , {Mmno) \ g, i 20 {abcde . fghij . klmno)2o 1 {abode . fghij . Iclmno)^^ 5 [abcde . fghij . 1clmno\ The theorems of the preceding section enable us to write down at once all of the imprimitive groups that have the above groups for heads except the second head of order 500. It is easily found that there are three groups that contain this head. The total number of these is found to be 55 and they may be written as follows : Order. No. 10368000 1 5184000 1 2, 3 2592000 1 2592000 2 1296000 1 2, 3 648000 1 48000 1 24000 1 2 12000 1 2 {abode) all {fghij) all {Mmno) all {afk . bgl . ohm . din . ejo) {af. bg.ch.di.ej) {abode) all {fghij) all {Mmno) all {afk . bgl . ohm . din . ejo) {{abode) all {fghij) all {Mmno) allj- pos {afk. bgl. ohm . din . ejo){af. bg .ch.di. ej){l, ab) I {abode) all {fghij) all {Mmno) all [ pos {afk . bgl . ohm . din . ejo) {abode) pos {fghij) pos {Mmno) pos + {abode) neg {fghij) neg {Mmno) neg {afk . bgl . ohm . din . ejo){af. bg .ch.di. ej) \ {abode) pos {fghij) pos {Mmno) pos + {abode) neg {fghij) neg {Mmno) neg \ {afk . bgl . ohm . din . ejo) {abode) pos {fghij) pos {Mmno) pos {afk . bgl . ohm . din . ejo) {af. bg .oh .di . ej ){l, ab .fg . M) {abode) pos {fghij) pos {Mmno) pos {afk . bgl. ohm . din . ejo) {abode)io {fghij)^ {Mmno)^^ {afk . bgl . ohm . din . ejo) {af. bg. ch.di. ej) {abcde)2t {fghij)io {Mmno)^^ {afk . bgl. ohm . din . ejo) \ {abode)^ {fghij )w {Mmno)^ \ pos {afk . bgl . chm . din . ejo) {af .bg .ch .di.ej) \ {abcde),o {fghij)^^ {klmno)^^ \ pos {afk . bgl . chm . din . ejo) {bced){af. bg .oh .di. ej) \ {abcde)^ {fghij)m {Mmno)^ \ pos {afk . bgl . chm . din . ejo) l\{abode)^ {fghij)^\ pos, {klmno)^^'] dim {afk. bgl. chm . din . ejo) {af. bg .ch.di. ej) 13 98 KuHN : On Imprimitive Substitution Groups. 1 2000 3,4 \ {ahcde^o {fg^lio - Qclmno)^ ] loo, 6 W^^ ■ ^9^ ■ chm . din . ejo) (of. bg.ch.di . ej){l, lo . mn) 6000 1 [{ {ahcde).iQ {fghij).^] poB, {Umno^^'] dim {afk . hgl. chm . din . ejo) 2 { {abcde)io {fghij)'" , (Tclmno\^ \ mo, 5 («/^'' ■ bgl . chm . din . ejo) 3, 4 {abcde\Q (fg^ijlio {Mmno\a {afh . bgl . chm . din . ejo) (af. bg .ch.di. ej)[l, bced . ghji . Imon) 3000 1 {abcde)iQ {fghij)i^ {Mmno)io {afh . bgl . chm . din . ejo) 2 ] {abcde)zo (fgMj)^^ {Hmnu)zo } 5, 5, 6 («/^ • ^9^ • chm . din . ejo) {af .bg .ch.di. ej) 3, 4 \{abcde)if, {fg^'b')i0 7 {klmno)io\ dim {afk. bgl . chm . din . ejo) {af. bg.ch . di . ej){l, be . cd) \{abcde)zo {fg^Vlzo {^^mno)2o\B, 5, 5 («/^ • ig^ • chm . din . ejo) \ {abcde)iQ {fghij)iQ, {klmno)iQ\ dim {afk . bgl . chm . din . ejo) \ {abcde)^^ {/g^ij )io {hlmno)^^\^^ 5, 5 {afk . bgl . chm. . din . ejo) {af. bg .ch . di •ej){l, bced . ghji . Imon) \ {abcde)io {fghij)io {hlmno)^^ [ 5, 5, 5 {afk . bgl . chm . din . ejo) {abcde)^ {fghij)^ {klmno)^ {afk . bgl . chm . din . ejo) {af. bg .ch .di. ej){l , be.cd . gj .hi .lo . mii) *{abcde .fghij.klmno).^^^ {afk. bgl . chm . din. ejo){af. bg .ch.di. ej) l\ {abcde)^^ {fg^y)io b, 5 . (^^'""0)30] 5, 1 («/^ • 'bgl ■ chm . din . ejo) {af. bg .ch . di . ej) {abcde)^ {fghij)^ {klmno)^ {afk . bgl . chm . din . ejo) *{abcde . fghij . klmno)i2Q (^/^ • bgl . chm . din . ejo) *{abcde . fghij . klmno)^ {afk . bgl . chm . din . ejo) {af. bg .ch . di . ej){l, ab .fg . kl) \_\{abcde)^ {fghij)Jf^^ s {klmno)^J\^^ 1 {afk .bgl. chm. din . ejo) \_\{ahcde)iQ {fghij)^^ \ dim, {Mmno)^^^^^ j {afk . bgl . chm . din . ejo) {af . bg .ch .di . ej){l, bced . ghji . Imon) *{abcde .fghij . klmno)^^ {afk . bgl . chm . din . ejo) [\ {abcde)iQ {fghij)xo \ dim, {klmno)^^^^ j {afk . bgl . chm . din . ejo) \ {abcde)^ {fohij)^ , {klmno)^\ 5, j {afk . bgl . chm . din . ejo) {af. bg .ch.di . ej){l, be.cd . gj .hi. lo. mn) 120 1 *{abcde.fghij . klmno)if) {afk .bgl.chm .din .ejo) {af. bg . ch.di. ej) 500 1 2 3, 4 750 1 2,3 720 1 600 1 375 1 360 1 2, 3 300 1 2, 3 180 1 150 1 2,3 75 1 60 1 2, 3 30 1 2, 3 15 1 Kuhn: On Imprimitive Substitution Groups. 99 ] (abcde)^ {fghij^ , (klmno\ \^^^{afk. hgl . chm . din . ejo) * {abode .fghij . ldmno\a {afk . bgl . chm . din . ejo) ■{abode . fghij . Jclmno)^ {afk . bgl . chm . din . ejo) {af .bg .oh.di.ej){l, boed . ghji . Imon) '{abode .fghij . Jdmno)^ {afk . bgl . chm . din . ejo) {abode .fghij . klmno)^ {afk . bgl . chm . din . ejo) {af. bg .oh.di. ej )(l, be.cd. gj .hi.lo . mn) {abode .fghij . Um.no)^ {afk . bgl . chm . din . ejo) * * Total, 55 Those groups marked with an * have also five systems of imprimitivity. The symmetric group and the alternating group of degree five can each be represented as an imprimitive group of degree fifteen. Hence identity occurs among the heads of the imprimitive groups of degree fifteen that have five sys- tems of imprimitivity. The transitive constituents of the other heads of these groups are cyclic groups of order three or symmetric groups of order six. And as in the preceding case each of these heads permits a cyclic interchange of its systems of intransitivity. It is found that there are nine groups that can be used for heads of imprimitive groups of degree fifteen that have five systems of imprimitivity. They are as follows : Order. 7776 {abc) all {def) all {ghi) all {jU) all {mno) all 3888 {{abc) all {def) all {ghi) all {jkl) all {mno) allf pos 486 \{abc) all {def) all {ghi) all [^jkl) all {mno) all j-g^ g^ g^ 3^ 3 243 {abc){def){ghi){jkl){mno) 162 {abc . dfe){def. gih){ghi .jlk){jkl . mon){ab . de . gh .jk . mn) 8 1 {abc . dfe){def . gih){ghi .jlk){jM . mon) 6 {abc . def . ghi .jkl . mno) all 3 {abc . def . ghi . jkl . mno) eye 1 Identity The theorems of the preceding section enable us to determine all the im- primitive groups that contain these heads except those whose substitutions inter- change their systems of imprimitivity according to either the symmetric or the 100 KuHN: On Imprimitive Substitution Orowps. alternating group of degree five. We note briefly the construction of the latter. Those that contain the head identity can be found at once by means of Dyck's theorem on the transitive representation of a given group.* The heads of orders 7776 and 6 are not contained in larger groups of the same degree that leave their systems of intransitivity unchanged. The groups that contain these heads and that correspond to (abcde) all or (ahcde) pos are determined then at once.f The groups that contain the head of order 3 and that correspond to (abcde) all or (abcde) pos are of order 360 or 180 and their factors of composition are 60, 3, 2 and 60, 3 respectively. The abstract groups with these factors of composition are known,J and hence we can at once find the corresponding imprimitive groups. The remaining groups (12 in number) may be easily found by tentative processes. I find in all 56 distinct groups that contain five systems of imprimitivity. Of these, 13 have also three systems of imprimitivity and so are found in the preceding list. Those which contain five systems of imprimitivity without also containing three systems are the following : Order No. 933120 1 (abc) all (def) SiW (ghi) all (jM) nil (mno) all (adgjm . behkn . c/ilo) (ad . be . cf) 466560 1 (abc) all (def) all (ghi) all (jht) all (mno) all (adgjm . behhn . cfilo)(adg . beh . cfi) 2, 3 \(abc) all (def) all (ghi) all (jM) all (mno) all}- pos (adgjm . behkn . cfilo)(ad . be . cf)(\, gh) 233280 1 \(abc) all (def) all (gJii) all (jkl) all (mno) all[ pos (adgjm . behkn . c/ilo) (adg . beh . cfi) 155520 1 (abc) all (def) all (ghi) all (jM) all (mno) all (adgjm . behkn . cfilo)(dgmj . ehnk .fioT) 77760 1 (abc) all (def) all (ghi) all (jkl) all (mno) all (adgjm . behkn . cfilo)(dm . gj .en .hk.fo . il) 2, 3 \ (abc) all (def) all (ghi) all (jkl) all (mno) all } pos (adgjm . behkn . cfilo)(dgmj . ehnk .fivl)(l, ab) * Mathematisohe Annalen, Vol. 32 (1883), p. 94. t Miller, Quarterly Journal of Mathematics, Vol. 28 (1895), p. 195. t Holder, Mathematisohe Annalen, Vol. 46 (1895), p. 417. KuHN : On Imprimitive Substitution Groups . 101 58 3 20 1 \ [abc) all (c?e/) all {ghi) all [jM) all {mno) all \ 3, 3. 3, 3, 3, {adgj'm . hehhn . cfilo){ad .he .cf) 38880 1 {abc) all {def) all {ghi) all {jhl) all (wino) all {adgjm . &eMn . c/V7o) 2, 3 \{ahc) all ((Ze/) all {gU) all (yA;Z) all (wino) allf pos {adgjm . hehhn . cfih){dm . gj .en.hk .fo . ^7)(l , ah) 29160 1 {{ahc) all {def) all {ghi) all (yM) all (wtio) allfg, ,. 3, 3, 3 {adgjm . hehhn . cfih){adg . heh .cfi) 2, 3 (ahc){def){ghi){jM){mno){adgjm . hehhn . cfih) {ad . he , c/)(l, ah .de . gh .jh . mn) 19440 1 \{ahc) all (cZe/) all {ghi) all (yH) all {mno) all} pos {adgjm . SeMra . c/i7o) 2 (a6c . dfe){def. gih){ghi .jlh){jhl . mon){ab .de.gh .jh . mn) {adgjm . hehhn . cfilo){ad . he . cf) 14580 1 {ahc)(de/){ghi){jhl){mno){adgjm . hehhn : c/ilo){adg .heh . cfi) 9720 1 \{ahc) all {def) all {ghi) all {jhl) all {mno) allf3_ 3^ 3, 3, 3 {adgjm . hehhn . cfilo){dgmj. ehnh .fiol) 2 {ahc . dfe){def . gih){ghi .jlh){jhl . mon){ah .de .gh .jh . mn) {adgjm . hehhn . cfilo){adg . heh . cfi) 3, 4 {abc . dfe){def. gih) {ghi .jlh){jhl . mon){adgjm . hehhn . cfilo) {ad . he . cf)(\, ah .de. gh .jh . mn) 4860 1 {{abc) all {def) all {ghi) all {jhl) all {mno) allfg, 3, 3, 3, 3 {adgjm . hehhn . cfilu){dm . gj .en.hh.fo. il) 2, 3 {ahc){def){ghi){jhl){mno){adgjm . hehhn . cfilo) {dgmj . ehnh . fiol){\, ah .de. gh .jh . mn) {ahc . dfe){def . gih){ghi .jlh){jhl . mon){adgjm . hehhn . cfilo) {adg . heh . cfi) {ahc . dfe){def . gih){ghi .jlh){jhl . mon){ab .de.gh .jh . mn) {adgjm . hehhn . cfilo){dgmj . ehnh . fiol) { {ahc) all {def) all {ghi) all {jhl) all {mno) all [3, 3, 3, 3, 3 {adgjm . hehhn . cfilo) {ahc){def){ghi){jhl){mno){adgjm . hehhn .cfilo) {dm . gj. en . hh .fo . il){l, ah.de. gh .jh . mn) {abc . dfe){def. gih){ghi .jlh)(jhl. mon){ah .de.gh .jh . mn) {adgjm . hehhn . cfilo){dm . gj .en.hh.fo. il) 3240 1 2430 1 2,3 1620 1 14 102 KuHN : On Imprimitive Substitution Groups. 1 6 20, 3 (abc . dfe){def. gih)[ghi . jlk){jkl . mon) [adgj'm . hehhn . cfilo) {dgmj . ehnk .fiul){l, ab.de. gh .jTc . mn) 1215 1 {abc)(def){ghi)(jkl){mno){adgj'm.behkn .c/ilo) 810 1 {abc.dfe){def.gih){ghi.jlk){jkl.mon){ab.de.gh.jk.mn) (adgjm . behkn . c/ilo) 2, 3 (abc . dfe){def. gih)(ghi .jlk){jhl . mon){adgjm . behkn . cfilo) {dm . gj . en hk . fo . iX){\i ab .de . gh .jk . mn) 405 1 [abc . dfe){def . gih){ghi . jlk){jkl . mim){adgjm . behkn. cfilo) 360 1 {abc . def. ghi .jkl . mno) eye (adgjm . behkn . cfilo) [def. gkn . hlo , ijm) 180 1 (abc . def. ghi .jkl. mno) eye {adgjm . behkn . cfilo) {aeh . hfi . cdg . mno) 120 1 {adgjm . behkn . cfilo){ad . be . cf) 60 1 {adgjm . behkn . cfilo){adg . beh . cfi) 43 Cornell University, June, 1901. Cornell University Library arY5 On imprimitive substitution groups 3 1924 032 189 684 olin.anx