90 



CHAPTER II. 



determining the structure of the Abelian group, we may therefore 

 suppose m > 1. 



111. We proceed to determine the substitution reciprocal to S 9 



m 



(i = 1,2,..., m). 



Supposing $ to be Abelian, we obtain the same result upon multi- 

 plying (p by tt that we obtain upon operating the substitution S upon 

 the two sets of indices. The identity of the two results is not 

 destroyed by operating the substitution S~ 1 upon the indices |,-i, vjn 

 (i = J ? . . , 9 m) of one set. The result obtained upon multiplying (p by p 

 and then applying the substitution S~ 1 upon the indices /i, r]n is 

 therefore identical with the result obtained by applying the substitution S 

 upon the indices , 2 , ^ 2 alone. Equating the two results , we find 



uOMja + ^-^ya) - ^i(.-J> + y/y 

 From this identity in the indices J,-^, i;^ 7 we find 



Hence the reciprocal of the Abelian substitution 75) is 



77) 



(t-l, 2, ...,m). 



When S~ l is operated upon the two sets of indices, cp must be 

 multiplied by l/^. Forming the relations expressing this fact, we 

 obtain the following conditions, together entirely equivalent to the 

 set of conditions 76): 



2 



78) l = 



<*ki 7k i \ 



7k i 



7k i 



Pa 



k i 



ft; a* ! 



= 



