THE ABELIAN LINEAR GROUP. 93 



to a n in S is not zero. Indeed, according as cr^-^O or y 1; =j=0, 

 we may take V = PIJ or P^jMj. Let S 1 replace |j by 



We can determine a substitution 5j_ derived from the above types 

 which shall replace j^ by o^, viz., 



Si = In, MiLi, a ft, 2, ^ #1, 2, y ia ft, m, lm 



where a and /? are determined by the conditions 



Hence S' = S t S", where S" is a new substitution of 6r which 

 leaves ^ fixed. Let S" replace ^ by 



.7 = 1 



For f* = 1, K n = 1, y n = 12 = y J2 = = I TO = yii 0> the relation 

 jR 12 = t a of 79) gives d n = 1 in the substitution S". The substitution 



will replace ^ by o 2 if we take 



Hence S n = ^ >S"", where $'" is a new substitution of G which 

 leaves ^ and ^ unaltered and thus has the form 



a-ii, ?; 



Cff . 



f ,r 



Applying the following relations of set 79), 



&.-0, fti-.O (< = 3, 4, ..., 2m), 

 n - / = M - n = (* = 2, 3, . . ., ). 



The relations between the coefficients a,-,, y,-^, /3,-y, ^ (*, j == 2, . . ., m) 

 of S'" are seen to be precisely those holding for a special Abelian 

 substitution on m 1 pairs of indices. Furthermore, 



S - FS' = F^S" = VS&S'", 



where F, 5 1? 5 2 were derived from the types of substitutions given 

 in the theorem. 



After m operations similar to that by which S"' was derived 

 from S, we reach a substitution which leaves fixed all the indices 



