I9I0.] AND ABSTRACT GROUPS. 313 



of G constitute a substitution group K which is simply isomorphic 

 to the quotient group of G with respect to the largest subgroup of 

 H- which is invariant under G. 



To see very clearly the relation between G and K it is perhaps 

 best to assume at first that they are simply isomorphic. We shall 

 represent by K^ the subgroup of G which corresponds to H in the 

 simple isomorphism between G and K. Hence K^ is also composed 

 of all the substitutions of K which omit a given letter a, as was 

 observed in the preceding section. As each of the co-sets is com- 

 posed of all the substitutions of K which replace a by a particular 

 letter we may name each of these co-sets by this particular letter. 

 In particular, K^ will be represented by a. If this is done it is 

 evident that the permutations of the augmented co-sets due to multi- 

 plying all these co-sets on the right by the same substitution will be 

 identical with this substitution. That is, the substitutions of K will 

 only be repeated by the permutations of the augmented right co-sets 

 when all of them are multiplied on the right by all the substitutions 

 of /v. That this arrangement was possible is a direct consequence 

 of the manner in which K was constructed, the details which we 

 gave are intended to exhibit more clearly how this may be done. 



We proceed to consider the substitution group K' which cor- 

 responds to the permutations of the augmented left co-sets when 

 these are multiplied successively on the left by all the substitutions 

 of K, in order. We again suppose that K^ corresponds to H and 

 that it is composed of all the substitutions of K which omit a. The 

 left co-sets are composed separately of all the substitutions of K 

 wdiich replace a particular letter by a. We shall name each of 

 these co-sets by this particular letter and hence H will again be 

 denoted by a. IMultiplying each one of these augmented left co-sets 

 on the left by the separate substitutions of K gives a permutation of 

 these co-sets represented by the inverse of the multiplying substitu- 

 tion. Hence we again obtain a repetition of all the substitutions of 

 K if we consider the permutations of the augmented left co-sets 

 when these are multiplied on the left by all the substitutions of K 

 in order. 



From the preceding paragraphs it results that the right and left 



