160 CHAPTER VII. 



172. Theorem. - - The order QU(W, p n ) of 0^(m, p n ) is, for m odd, 



2) n(m 3) _ 



and, for m even, 



[ ( 

 pn(rn-i) qi 2 p"\ 2 



2 n 



?'s - - or -f according as ft = 1 or i>, 

 = 1 according to the form 4:1 + 1 of p n . 



We notice first that the number of substitutions S, S', . . . of 

 Ofi( m > P") which leave | t fixed is Q^(m 1, jp n ). In fact ? they have 

 a n = 1, cc 12 = 13 = = i m = 0, and therefore by 146) for j = 1, 



*! = (Jc-2,.. ., w). 



Hence they belong to the group 0^(tn \,p n ) leaving invariant 



i! + iI+---+&- 1 +ft&. 



Let T be a general substitution of 0^ (m ; j9 w ) and let it replace t by 



where, by 145) for j = l, 



147) aj, + J 2 + - - + i w -i + y J W - 1. 



The 0^(^1,^") substitutions TS, TS', . . . and no others of the 

 group will replace | by Oj. If, therefore, P^(m, p n } denote the 

 number of distinct linear functions o l by which the substitutions of 

 OP ( m > .P") can replace ^ , we have for the order of the latter group, 



This recursion formula gives 



^(m, p n ) = P^(m,p n ) P^m - 1, p n ) . . . P^,p}, 



since the identity is the only substitution of determinant unity on 

 one index which leaves ftgj, invariant, so that Q(l,j) w ) = l. 



It will be shown in 174 180 that P^(k,p n ) equals the number 

 of sets of solutions in the GrF[p n ~\ of the equation 



and hence, by 65 66, P^(k,p n ) = 



k_ (jk__ 1 \ 



^(*-Dq: 2/va ) (j. even ) 

 * i 



p n(k-l) e ~T~^n(t-l)/2 (^ odd) 



the upper signs holding if fi = 1, the lower signs if ft = v, and 



denoting +1 or 1 according as 1 is a square or a not -square 



