I9I0.] AND ABSTRACT GROUPS. 311 



Hence ^ts order is a power of p. As H and So'^HS^, have a sub- 

 group of index p in common and as such a subgroup is always invar- 

 iant in a group whose order is a power of p, it results that these 

 common operators form an invariant subgroup of K. This is im- 

 possible unless the order of this common subgroup is unity and 

 hence K, cannot involve a transitive constituent of order p unless 

 the order of K^ is p.^ 



When K^ involves a transitive constituent of degree n^, Hand 6"o~^ 

 HSn have a subgroup of index n^ in common and vice versa as was 

 proved above. It is also known that G involves a multiple of nji 

 operators which transform H into a group having exactly a sub- 

 group of index n^ in common with H. It is evident from the above 

 that there are exactly nji such operators for every transitive constit- 

 uent of degree w^ in K^. Hence we have the following theorem: // 

 G involves knji operators ivhich transform H into a group having 

 exactly a subgroup of index n^ in common zvith H then must K^ 

 have exactly k transitive constituents of degree n^ and vice versa. 

 In particular, K is multiply transitive when K^ is transitive and of 

 degree p-i. Hence the necessary and sufficient condition that K 

 is multiply transitive is that G contains an operator which trans- 

 forms H into a group which has exactly a subgroup of index g/h 

 — I in common with H, g and h being the orders of G and H re- 

 spectively. This theorem gives meaning in certain cases to the 

 theorem, if a subgroup //^ of a given group G has exactly p opera- 

 tors in common with a conjugate of a subgroup Ho of G, then the 

 number of the operators of G which transform H„ into subgroups 

 having exactly p operators in common with H^ is khji^-^p, h^ and 

 hn being the orders of H^ and Hr. respectively. The given theorem 

 gives a meaning to k whenever H-^ and Hn belong to the same system 

 of conjugates. 



From the main theorem of the preceding paragraph it is easy to 

 obtain the degrees of all the systems of intransitivity of K^. To 

 obtain the orders of the transitive constituents of K^ it is only 

 necessary to observe that if S^HS^~'^ has a subgroup of index n^ 

 in common with H and if the largest invariant subgroup of H con- 

 tained in this common subgroup is of index m^ under H then the 



^Proceedings of the London Mathcmaiical Society, vol. 28 (1897), p. 536. 



