The non-regular transitive Substitution groups 

 whose Order is the produot of tliree unequal prime numbers/) 



By 



G. A. Miller at Paris. 



We shall represent the tliree prime numbers by p, q, r and 

 assume that iJ> q> r. Since the order of a transitive group is 

 a multiple of its degree and all the groups in question contain 

 an invariant (selfconjugate) subgroup of order p^) the degree of 

 these groups must be p, pr, or ^g. We shall examine all the 

 possible groups for these three degrees in the given order. 



§ 1- 

 The transitive groups of degree p and of order pqr. 



The largest group ( H) that transforms the subgroup of order 

 p into itself transforms its substitutions according to the cyclical 

 group of Order p — 1, for p has primitive roots. Hence it is only 

 necessary to consider the subgroups of order qr which are eontained 

 in this cyclical group. 



Since a cyclical group has one and only one subgroup cor- 

 responding to each divisor of its order, the given group of order 

 p — 1 has one subgroup of order qr, when pi — 1 is divisible by 

 qr. If this condition is fulfiUed, jBr(the metacyclic group) has one 

 and only one subgroup of order pqr ^). We shall represent this 

 group by G^ . It contains p> — 1 substitutions of order p, p [q — 1) 

 of Order q, p {r — 1) of order r, and p {qr -\~1 — q — r) of order qr. 



^) The regulär groups of this order were determined by Cole and Glover 

 (American Journal of Mathematics, vol. 15, pp. 215—220) and by Holder 

 (Mathematische Annalen, vol. 4.3, pp. 361 — 371). 



^) Cf. Frobenius, Sitzungsberichte der Akademie zu Berlin, 1893, 

 I, p. 343. 



^) Cf. Netto, Substitutionentheorie, p. 151. 



