166 CHAPTER VII. 



According to the definition of a, /3, ;/, d in the above cases, the 

 expressions 



belong to the 6r.F[^) n ]. The above conditions then give 



so that O^s must be the special substitution Q^s defined in 181 

 Any orthogonal substitution 02,3 not of the form 2,3; and therefore 

 not of the form .X, will extend Oi(3,^9 n ) to a larger subgroup 6r of 

 Oj(3, ^ n ). The order of 6r is therefore at least p n (p 2n 1). From 

 the remarks at the end of 177, it follows that 6r has exactly this 

 order and hence coincides with 1 (3, p^. 



179. We proceed to the proof that, if -- 1 be a square i* in 

 the GrF[p n '], the group 1 (3,p w ) is generated by the substitutions 

 Of/ together with E 12S if p n = 5. If p n > 5, there exist ( 64) marks 

 /3 and r in the GF[p n ~\ such that 



1+ /*_, + 0, T + 0). 



Then the product 



^ i L wn ich is an 6)2,3, transforms _ fj into . ^ 

 Furthermore, 



ri /jnri ^n_ri /s (! + 

 Lo i J Lo i J Lo i 



Since /3 =f= 0, we can.( 64) find marks a^ and a 2 in the field such 

 that /3(a 2 + |) = x, where % is an arbitrary mark =j= 0. Also 



r o il- ] fi -*-ir o 11 ri on r o n 1)0 

 L-i oJ Lo iJUi oJ = L J' L-i oJ 5 ^' 



Hence, if p n > 5, we have reached from the O"'/ the substitutions 



[J J], [1 J] (x arbitrary). 



By 100 and 108, these substitutions generate the group LF(2,p n \ 

 Hence the OJ/ from which they were derived generate the isomorphic 



