132 CHAPTER V. 



For m = 2, we have by 117) and 121), when A 1 =A 2 = 1, 



Inversely, every substitution satisfying these relations is seen to leave 



If " 1 + i? " 1 absolutely invariant. Every such substitution is the 

 product of a substitution 



122) 

 by one of the p* + 1 distinct substitutions 



The number of distinct substitutions 122) is (jp 8 * l)j>*. Indeed, for 

 the #*+! values of 12 for which *=*!, we must have a n = 0; 

 while for each of the remaining (p 2s p s 1) values of a 12 in the 

 there exist p s -}- 1 solutions in the field of 



for, the second member belongs to the GF\^p^\ and is therefore the 

 p s -\-\ power of some mark in the G-F[p* s ]. But 



The group of the substitutions 122) has an invariant subgroup 

 of order 1 or 2, according as p = 2 or p > 2, generated by the 

 substitution 



The quotient group (obtained concretely by taking the substitutions 122) 

 fractionally) is, by 137 138, simply isomorphic with the group of 

 linear fractional substitutions of determinant unity in the GrF[p s ~]. 

 By 109, it is a simple group when p s > 3. 



Corollary. Every binary hyperorthogonal substitution in the 

 GF[p 2s ] taken fractionally may be given the form 



-^- B \ 

 -X A*i 



of determinant a mark of the GF[p'] y where J., B belong to the 



Indeed, since D p * +1 = l, we may set D = R p * 1 , ^belonging to 

 the G-F[p* 8 ]. The fractional binary hyperabelian substitution becomes 



