72 G. A. Miller. 



of Order j; in H-^ into a power which belongs to exponent p — 1 

 ■\vith respect to niodulus p and does not interchange the cycles of 

 these substitutions. we obtain a group of order p^q {p — 1) that 

 contains Ä> as an invariant subgroup. This group must, therefore. 

 be B'. 



The groups in question must be subgroups of H^ and cor- 

 respond to a group of order r in the group which is isomorphic 

 to H^ with respect to the given invariant subgroup of order pq. 

 The Order of this isomorphic group is j) [p — 1). We have proved 

 that it is isomorphic to a cyclical group of order p — 1 with 

 respect to its invariant subgroup of order p. Hence there is one 

 and only one group of the required type, whenever p — 1 is 

 divisible by qr. We shall denote this group by G^. 



Gi contains an intransitive invariant subgroup of order pr 

 which may be constructed by establishing a simple isomorphism 

 between q transitive groups of degree p and order pr. Its other 

 substitutions not found in Ifo are all of order qr. It remains only 

 to examine the case when the invariant subgroup of order pq is 

 cyclical. 



The substitutions of these groups, which are not contained 

 in the given invariant subgroup of order 2)q {Hci/c), must transform 

 the substitutions of Hct/v. into powers which belong to the ex- 

 ponent ;•. modulus p. To each group correspond r — 1 dififerent 

 powers. Since the congruence 



ic" = 1 (mod 2)q) , p> q> r 



has one root. when neither p — 1 nor q — 1 is di%asible by r. 

 r roots, when either p) — 1 or (^ — 1 is divisible by r, r^ roots, 

 when both p — 1 and q — 1 are di\'isible by r *), and since the root 

 unity clearly does not correspond to a group ; there is one group 

 of the required type, when either p — 1 ov q — 1 is divisible by ;•, 

 and there are r -f- 1 groups, when both p — 1 and q — 1 are 

 divisible by r. 



These groups are generated by Heye, and substitutions of 

 order r which transform any Substitution of order pq in Heye, into 

 one of the required powers. The substitutions of order )• may 



') Cf. Gaus.*. Disqiiisitiones arithmeticae. Sectio iii. Art. 92. 



