106 



CHAPTER II. 



123. Theorem. - - Within the special Abelian group on 2m indices 

 in the GF[p n ], p > 2, every substitution S, sucli that S 2 = T, is conju- 

 gate with M = M^Mz . . . M m . 1 ) 



Taking as S the general substitution 75), whose reciprocal is 

 given by 77) for ft == 1, the condition S = S~* T is seen to require 



.V = - d J iy V*i - YJ iy Pv = Pjt ft j = 1, . . .,' w). 

 The matrix of coefficients of the general S is therefore 



8 = 



'12 



Vim 



72m 



-"1m 7lm U 2m 72 m 



ftm tflm /^2m "~2??i- $ 



subject to the special Abelian conditions. 



Take first m = 1. Then 5 has the form 



It is conjugate with a similar substitution in which a u = 0. In fact, 

 if /3 n =|= 0, the transformed of S by L^ % replaces r^ by 



in which the coefficient of % may be made zero by choice of L If 

 fti == ^ ^11 H 55 ^ we fi rs ^ transform S by Jfefj and then proceed as 

 before. If /3 n = y u = 0, we first transform /S by L[, i and obtain a 

 substitution which replaces ^ by 2JLa 11 ij 1 n^i; so that the new 



Ai + O. 



With a lt = 0, S takes the form 



and is the transformed of M by the special Abelian substitution 



(f 



Indeed, by 64, there exist solutions in the 6rJF[p 71 ], ^> > 2, of 



To prove the theorem by induction for m pairs of indices, we 

 assume it true for t pairs of indices t < m. 



1) For the number of conjugates see Ex. 8, end of Ch. VIII. 



