190 CHAPTER VII. 



In the latter case, we transform $ 2 by Ol^fy'e and require that 



$2 = afts + bPm2 + eft, + d/3 52 + e/3 62 = 0, 

 /Jisa/k + a/ki-O. 



the second condition becomes an identity and the first takes the form 



e &2 = 6 & P = ISrPn Pm2* 

 Psi 



The further condition a 2 + 6 2 -f c 2 -f d 2 -f e 2 = 1 then becomes 



177) 

 Since ft + PL + l, /Sf. + ^+^+ft^i-l, i* follows that 



178) 7 + f- 1 + ' 



The coefficient of & 2 is zero for at most two values of /3 62 . In view 

 of 178), these two values of /3 62 can be avoided, if ^) w > 3, by an 

 earlier transformation of $ 2 by 456 , an operation not affecting the 

 previous argument. Also the coefficient of d 2 is not zero. Hence 177) 

 has solutions d, I in the field. The conditions /3g 2 = fi' 6l = can thus 

 be satisfied. 



For p=3, 178) requires ^ = 1. The coefficient of & 2 in 177) 

 is then zero only when /3 =f= 0. If |8 =(= 0, we can determine a and 6 

 (each =J= 0) such that 



Since a 2 = 6 2 = 1, ^ = 1, the remaining conditions become 



These are satisfied modulo 3 by taking c = /3 42 , d = /3 52 , e = 0. 



193. We have thus reached in G a substitution Z which leaves 

 fixed 6 , 7 , . . ., m _i and which is not the identity. If 



X = G! C 2 C 3 (7 4 (7 5 (7 m , 



we obtain from it the substitution C^C/g as at the beginning of 191. 

 From the known structure of the subgroup 0^(6,p n ), it follows that 

 G contains all the substitutions of this subgroup. Transforming these 

 by suitable even substitutions on the t -, we obtain all the generators 

 of 0(w,# n ), with which G therefore coincides. 



194. In stating our results concerning the structure of the 

 orthogonal groups on m =f= 4 indices, we introduce permanent 

 notations for the simple groups reached. For the first orthogonal 



