THE ABELIAN LINEAR GROUP. 



107 



If KH = 0i t = yn = #1,- = (^ = 2, . . ., m), then $ = $!$', where 

 $! affects only | 1? ^ and is therefore conjugate with M 19 and where 

 $1 affects only /, ^,- (^ = 2, . . ., m) and is, by assumption, conjugate 

 with M 2 M S . . . M m . Hence $ is conjugate with M 1 M^M^ ... Jf m 

 within >S A (2 m, j ra ) . 



In the contrary case, 8 is conjugate (by 120) with one of the 

 two substitutions Z 17 Z 2 . We consider the following three cases. 



Case a). If Z 1? with d =(=.0, be of the form 8 above, the Abelian 

 conditions give 



/j J^ f\ / * G) ~1\ 



Hence Zj = Z^ Zj_ f , where Z^ has the form 



Im = 



while Z" affects only | f , ^ f (^ = 2, . . ., m 1) and is, by assumption, 

 transformed into M$Ms . . . Mm i by some special Abelian substitution 

 affecting only the same indices. We proceed to prove that Z' x may 

 be transformed into MiM m by a special Abelian substitution on the 

 indices 1, T?I, | m , ^ m . The proposition that ^ is conjugate with 

 MiM 2 ... J4* under SA(2m,p n ) will then follow. 



If ^ = ^=^=0 in Zj, we transform it by N^ mi i and get 



This is of the form !.{, but has y u 4= 0. Next, if y u = 0, /J u 4= 0, 

 we transform Z^ by M^M m I m ^ and get a substitution of the form 

 Zj in which, however, y u =}= 0, /3 11 = 0. We may therefore assume 

 that y n =|= in Z' x . Transforming it by Li^L'it where l = a 

 we get a substitution of the form 











dy u 







