270 CHAPTER XII. 



respect to the GF[p~\. By 241 , G p m is transformed into itself 



only by substitutions of the form V = ( ' _-,] Since F transforms 



\o, K V 



& into S&JL, a further condition is that a 2 1 and A should run 

 simultaneously through the series of marks of the [A 1; . . ., AJ. 

 Suppose that there are in the GF[p n ] exactly e marks f 17 . . ., s e 

 such that [A t , . . ., AJ = [? AI, . . ., 4 AJ. Then, according as p > 2; 



^) = 2, the ^ substitutions 



where /3 ranges over the GF\s\, constitute the largest subgroup H 

 of G M (s) under which G p m is self -conjugate. But the multipliers K 

 of the additive -group [Aj, . . ., A m J are ( 70) the marks K =j= of the 

 multiplier GF\_yF\, k being a divisor of m and w. It remains to 

 distinguish which of them are squares of marks Si of the GF[p n ~\. 

 For the respective cases 



p > 2 with n/Jc even, p > 2 with w/& odd; p = 2, 

 there are ( 62) exactly e = (2, 1; !)(#* 1) marks /, so that If is 

 of order Hence G p m is one of a system of 



J ; 1 1, J; 1 



(*--l)-T-ftl;l)(p-l) 



conjugate subgroups of G M (s)- Here the value of ~k depends on the 

 individual 6Jy chosen. Given ^, the number of the corresponding 

 sets of G p m follows from 71. 



250. Non- commutative subgroups of the s + 1 conjugate 6r$_i) . 



2; 1 



It suffices to study the group G given by ^ = oo. It is composed 

 of the substitutions 



For a given mark a =[= ? ^ and /3 run simultaneously through the 

 series of marks of the GF\s\. A rectangular array of the substitu- 

 tions of G may be formed by taking as the first row the substitu- 

 tions Sp, which form the self -conjugate subgroup 6r, , and as right- 

 hand multipliers the substitutions P a = ( -\ of the cyclic 6r^i 0) . 



\0, a J yr^ 



In any subgroup G' of G the totality of substitutions of period p 

 give rise to a commutative group G p m of substitutions ft, where A 

 ranges over an additive -group [A 1; . . ., AJ. Hence 6iyn is self- 

 conjugate under 6r'. A rectangular array of the substitutions of G' 



