98 CHAPTER II. 



For p = 2, a substitution $ of G = SA(2m, p n ) is commutative 

 with every L^ i and every Z/, % only when S is the identity. Proceed- 

 ing as in 116, we find that a self -conjugate subgroup J of 6r, 

 which contains a substitution S =%= I, will contain either a substitution 

 of the form 81) with a and /5 not both zero or else a substitution S' 9 

 of the form 84) in which y u =j= 0. 



We next prove that J contains . either Li t 2. (h =4= 0) or else 

 -#i, 2, i 2, i. For d = 0, 2 = Zi, 7ll . For d 4= 0, we transform 2 by 

 a suitable TI ? * T 2? ^ and obtain the substitution Zi, i L 2 , i. Hence J 

 contains 1 ) t 



VI, 2, 1 -L/l, 1 -^2, 1 Vl, 2, 1 = -l-n, 2, 1 -^2, 1- 



For a = 0, 81) becomes Z 2 , /s, so that we reach Li,p in J. If 

 /3 = 0, 81) becomes JYi f 2 , a, so that, by 82), J contains i lj _ a a. 

 Finally, if 4= 0, /3 4= 6,' the transformed of 81) by 1^1*^ gives 

 the substitution 



S = Si + ^ 2 , |; = |j -f 

 In the G-F[2 a ], we may take 



ft -/J- 1/1, A = ->-*, 



when the last substitution becomes JV^ 2, i -^2, i 



Having a substitution Zi,/ (A =)== 0), J will coincide with Gr. 

 Indeed, TI^ transforms L^z into L^jn?. Since every mark of the 

 field is a square, we reach L^ a , 6 arbitrary. Then, as at the end 

 of case a) of 116, J contains every L^ OJ M iy N^^ and hence 

 coincides with G. 



There remains the case in which J contains N^ 2, 1^2,1- Then 

 J will contain all the products, two at a time, of the substitutions 



85) L f , i, M h N^ ! (i, j=l,2,.. ., m; 



Indeed, if i and j be any two distinct integers ^ m, J contains 



I, i j t j t i j t i j t j, i = f) i y ? i , 

 MT*L it i L it ! M t - L t> ! Lj, , = L>, , M { , (L,- , M^ = M t L^, 



L^ ! L it ! . L,, ! Mi = Lj, 1 Mi - Jf, Zy, !, Jft L it x ' Zy, ! Jf; = Jf, Jf,, 



Our statement is therefore proved if m = 2. If m > 2, let i, j, Jc be 

 any three distinct integers <; m. Then J contains 



N{ } j } i L it i L{ } i L^ i = N.; t j t i L^ i = LJC } i N^ ^ i , 

 N itit i i/, i Z/, i Jf* = ^,,- iM k = M k Nf t/1 1. 



1) This relation follows from 83), if p = 2, by taking i = 1, j = 2, p = 1. 



