164 CHAPTER VII. 



Hence at least one set of solutions of 149) will make 1 a^ 2 a 

 square or zero. If it be a square, the theorem follows from 174. 

 If it be zero, we have by 148), 



/y' 2 _ JL '2 



"i = ^ a s 

 Hence a{ = as = 0, 2 2 = 1; so that we may take $ = 0ij2 a - 



177. For the case 1 ) la square in the CrF[p n ~\ and p = l, it 

 follows from 178179 that 1 (3,jp w ) contains a subgroup of order 

 at least p n (p 2n 1) generated by the substitutions Of/, together with 

 -^123 if # n = 5 > a11 of determinant + 1. But, by 172, the order of 

 Oi(S,p n ) is P 1 (3,jp n )P 1 (2,^ w ). Here P^, p n ) =p n -l, being the 

 number of functions 



by which the substitutions of 1 (2,jp w ) can replace |j. Also 



In fact, if a substitution of 1 (3,p w ) replace ^ by 



then i = ii 



150) J + J -f- | == 1. 



By 66, this equation has yP n -\- p n sets of solutions in the 

 -1 being a square. The order of Oj(3, jp n ) is thus 

 (p 2w + p n )(j? n 1). From the two results it follows that this number 

 equals the order of O x (3, p n ) and that for every set of solutions 

 of 150) there exists a substitution of O x (3, p n \ derived from Of/ 

 and JR, which replaces ^ by c^. 



178. Theorem. The first orthogonal group 1 (3 ) p n ) contains a 

 subgroup 01(3,j) w ) Twloedrically isomorphic with the group LF(2,p n ') 

 of linear fractional substitutions of determinant unity. 



Let 1 = ^ 2 , so that i belongs to the G-F[p n '] if and only if 

 1 be a square in that field. Introduce in place of | 1; | 2 , 8 the 

 new indices 



%=-*!i> %=6s *8s> %=la+*S8^ 

 so that 



1) For a more direct treatment of this case, but one involving considerable 

 calculation, see Amer. Journal, vol.21, pp. 202 204, in which the proof of 

 Jordan, Traite, pp. 164 166, for n = 1, is corrected and generalized. 



