THE ABELIAN LINEAR GROUP. 



103 



G-F[p n ], p > 2, on t < m pairs of indices is conjugate within the 

 group SA(2t 7 p n ) with one of the substitutions T r (r <; ) and proceed 

 to prove that a like result holds for m pairs of indices. In view of 

 120, we may suppose that S has one of the three forms Z 1? Z 2 

 or S lf the latter having lf = yu ^ ft* = \* = (i = 2, . . ., m). 

 An $ of the form S is evidently a product T^ + iS*, where 82 affects 

 only the m 1 sets of indices | 2 , ?y 2 , . . ., m , ?? m . By hypothesis, 

 5 2 is conjugate with one of the products, J, J 2j _i, T 2> _i T 3j _i, . . ., 

 Tg,__i ^3, i T m , i. Hence an $ of the form $ t is conjugate with 

 some I r (r = 1, 2, . . ., m). We proceed to consider Z x and Z 2 in 

 the following three cases. 



Case a), d =j= in X x . Then /S' has the form 



The Abelian conditions 79) give at once 



Kim = Pirn = 7im = dim (* = 2, . . ., W 1), flu + K mm = 0. 



Hence I x = !.( I/, where Z" affects only the indices 



g/, ^- (' = 2, . . ., m - 1), 



while Z^ affects only |i, ^, m , ^ m , viz., 



n * 



n d 



d - u 

 010 - n . 



By hypothesis Z^' is conjugate with some product of the 



In order to make the induction from one to two pairs of indices, 



J.J-L V^A VtV/J. VV^ AAMWBbV WJ-i 



we must prove that 

 T w? _ i. Transforming 



n _L(y# 



u -f 



d 



1 



induction from one to two pairs of indices, 

 ^ is conjugate with a product of TI,_I and 

 Z' t by Ci,m,a, we obtain the substitution 















-a n -<5d 







