92 CHAPTER II. 



S - U r T, 



where T is a new Abelian substitution not altering cp and hence in 

 the special Abelian group. Since r may be any one of the integers 

 1, 2 7 . . ., p n 1, the order of GA(2m, p n ) is p n 1 times the order 

 SA[2m,p n ~] of the group SA(2m, p n \ 



Let a, /?,... be the prime factors whose product gives JM W 1. 

 Let ^L, J., A tt p, . . ., -4.p_i = SA(2m, p n ) be the groups formed by 

 the combination of the substitutions of SA(2m, p 7 *) with 



Z7, Z7, 7<*, . . ., Z7*"- 1 = / 

 respectively. Evidently these groups have the respective orders 



(p n - 1) SA [2 m, p n ~] , ~- (p n -l)SA[2m, p n ] ,, 



while each is self - conjugate under A = 



114. Theorem. - - The group SA(2m, p n ) is generated by the 

 substitutions x 



where i, j = 1, 2 ; . . . ? m; i =j=^'; aw^ where k is an arbitrary mark of 

 the GF[p n ~\. Every substitution of the group lias determinant unity. 



From these substitutions leaving cp absolutely invariant, we 

 obtain other simple substitutions of SA(2m, yf) = G as follows: 



Let /S be any substitution of 6r and let it replace ^ by 



yiy not a11 zer ]- 



We can set S = F$', where T^is derived from the above substitutions 

 and S' is a substitution of G in which the coefficient corresponding 



1) In the expression for each substitution we omit the indices not altered. 

 For example, MI alters only the two indices v\i and | . 



