Xon-regular transitive Substitution irroups. 69 



The number of groups of this type, which exist for a given 

 valiie of p, is clearly eqiial to the number of pairs of unequal 

 prime factors contained in jj.» — 1. Hence such groups exist always, 

 when 2' is larger than 5 and p — 1 is not a power of 2. The first 

 value of j) for which there is more than one such group is 31. 

 In this case there are three groups. Their Orders are 186, 310, 

 and 465 respectively. 



§ 2. 

 The transitive groups of degree pr and of order pcp: 



The invariant subgroup (Ä, )of order jjg ') must be intransitive, 

 for its Order is not a multiple of its degree. Since its Systems 

 of intransitivity are permuted according to a transitive group of 

 Order r, their number must be r. H^ may, therefore, be formed by 

 establishing a simple isomorphism between r transitive groups of 

 Order j^'l- -^s the latter can exist only when j) — 1 is divisible 

 by q. there can be no transitive groups of degree j;r and order jjqr 

 unless this condition is satisfied. In what follows we shall suppose 

 that it is satisfied. 



If we add to B^ a Substitution (t) which merely interchanges 

 its r Systems of intransitivity, we obtain a group (G.y) of the 

 required type. Go contains the cyclical group of order p»-. Each 

 one of its other substitutions transforms the substitutions of the 

 subgroup of Order p into one of the q — 1 powers w^hich belong 

 to the exponent q, modulus j;. Those which are not contained 

 in H^ are of order qr. 



"When p — 1 is divisible by qr, we may constriict a second 

 group (Gg) of the required degree by using, instead of t, the 

 Substitution obtained by multiplying into t a Substitution which 

 transforms the substitutions of the subgroup of order p into some 

 one of the r — 1 powers which belong to the exponent r, modulus p. 

 G^ contains the non-cyclical transitive group of order pr. Those 

 of its other substitutions which are not found in H^ are of order qr. 



The other groups which may be constructed in the same 

 manner as G^ are conjugate to it with respect to substitutions 

 which merely interchange the Systems of Z/^. Hence there are 

 two groups of degree pr and order pqr, whenever p — 1 is divisible 



M Cf. Frobenius, loc. cit. 



k 



