308 MILLER— SUBSTITUTION GROUP PROPERTIES [J«ne 4. 



The sets HS^, S^H (a = 2, 3, . . ., p) are known as co-sets of G 

 as regards H. Galois called attention to the importance of the case 

 when each HS^^SJ-i for every operator of H. That is, for every 

 operator of H in the first member there is some operator of H in 

 the second member so that this equation may be satisfied. The 

 necessary and sufficient condition that H is invariant under G is 

 that such equations may be established for every value of a from 



2 to p. 



The co-sets HS^, S^H will be called right and left co-sets re- 

 spectively. The totality of these co-sets is independent of the choice 

 of the operators ^S'o, So, . . ., S ; that is, there is only one category 

 of right co-set, and only one of left co-sets as regards any given 

 subgroup. Each left co-set is composed of the inverses of all the 

 operators in a right co-set and vice versa. Hence we may say that 

 the necessary and sufficient condition that a subgroup H is invariant 

 under a group G is that every right co-set of G as regards H is iden- 

 tical with some left co-set as regards H, or that every left co-set is 

 identical with some such right co-set. In other words, the necessary 

 and sufficient condition that H is invariant under G is that the num- 

 ber of distinct co-sets to which H gives rise is equal to its index 

 under G diminished by unity, or that the totality of its right co-sets 

 is identical with the totality of its left co-sets. This theorem is 

 included in the theorem that the necessary and sufficient condition 

 that H is invariant under a right co-set is that this co-set is identical 

 with some left co-set, or vice versa. 



With the extreme case in which each right co-set is identical with 

 some left co-set we may contrast the other extreme case when each 

 right co-set has some operator in common with every left co-set. It 

 has been seen that H is invariant in the first case and we shall see 

 that it gives rise to a multiply transitive substitution group (iv)* in 

 the second case, excluding the trivial case when the order of K is 2. 

 In the proof of this theorem it will at first be assumed that H is 

 neither invariant under G nor does it involve any invariant subgroup 

 of G besides the identity. That is, we shall at first assume that K 

 is a transitive substitution group which is simply isomorphic with G. 



■• Dyck, Mathcmatischc Annalcn, vol. 22 (1883), p. 91. 



