THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 133 



The group may be transformed into the group of all linear fractional 

 substitutions in the GF[p s ] (see 138, 137, corollary II). 



145. For m general, let S be an arbitrary substitution of 



By 139, its inverse is obtained by replacing a t -j by a?* . Hence the 

 relations 117) and 118), for A 2 = 1, when written for the inverse S~ l , 



771 



give the equivalent set of conditions for the invariance of 

 123) 



124) ^ X = U * - 1, . . ., ;>' -H *> 



?' = ! 



By 146, the number of distinct linear functions 



by which the substitutions of G- m , p , s can replace t is the number 

 Pro, P.* of distinct sets of solutions in the G-F[p 2s ~\ of the equation 



125) 



Let T be a substitution of the group which replaces fjj by a 

 definite function f v Then, if Z, Z', . . . denote all of the Q m , PlS sub- 

 stitutions of the group which leave ^ fixed, the products TZ, TZ', . . . 

 and no other substitutions of the group will replace t by f v Hence 

 the order Q W)2 ,, S of the group G m , py * is 



Q ^^ O P 



772j J), S ~ jjWli P) ^ ffl) Pi S * 



But the substitutions Z, Z f , . . ., have 



n -l, !, ( = 2,...,w). 



Hence by 124), for j = 1, we have 



Hence Z ? Z', . . . are substitutions of the group Gr m i t p,* on 

 indices | 2 , . . ., | OT , so that ^, p , s - Q m _i,^,. Hence, since 



