312 MILLER— SUBSTITUTION GROUP PROPERTIES [J"ne 4, 



corresponding transitive constituent in A\ is of order m^ and of 

 degree n^. The fact that its order is m^ results directly from the 

 facts that the ii^ right co-sets which involve all the operators of 

 HSH have the property that each of them is unchanged when mul- 

 tiplied on the right by any one of the operators of this invariant 

 subgroup. ^Moreover, it is evident that the identity of each transi- 

 tive constituent of K^ must correspond to an invariant subgroup of 

 K^. Hence it results that // H^, i7„, . . ., H^ be any complete set 

 of conjugate maximal subgroups of G then each index under any 

 one of these subgroups as regards the largest invariant snibgroup-. 

 zi'hich it has in common unth any other is divisible by the same prime 

 numbers for every possible pair in the set of conjugate subgroups. 



As a special case of the theorems proved in the preceding para- 

 graph we have the following: The necessary and sufficient condi- 

 tion that K-^ is composed of simply isomorphic transitive constitu- 

 ents is that every invariant subgroup of H which occurs in one of 

 its conjugates under G occurs also in all of these conjugates. This 

 condition must clearly be fulfilled when K^ is transitive ; that is, in 

 this case H cannot have an invariant subgroup in common with any 

 one of its conjugates unless this subgroup is also invariant under G. 

 This theorem exhibits an interesting abstract group property which 

 corresponds to the property that K^ is composed of simply isomor- 

 phic transitive constituents and with those mentioned above estab- 

 lishes more completely the principle of duality as regards substitu- 

 tion groups and abstract groups. 

 § 3. Transitive representation as regards right and left co-sets. 



Since both the right and the left co-sets of G are determined by 

 the subgroup H each of these two categories of co-sets together with 

 H forms a totality whose elements are permuted among themselves 

 when the former are multiplied on the right and the latter on the 

 left by operators of G. The p sets obtained by adding H to the 

 right co-sets will be called the augmented right co-sets. Similarly 

 we shall use the term augmented left co-sets for the p sets obtained 

 by adding H to the left co-sets. It is known, and also evident, that 

 the permutations among themselves obtained by multiplying the aug- 

 mented right co-sets on the right successively by all the operators 



