THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 135 



and which is generated ~by tJie following substitutions [in which only the 

 indices altered are written]: 



additional generator ~being necessary if p g =2, m^ 3, vie., 



TF: 



= i x + 1 6, + P| s [-P = 1+ 1 (mod 2)]. 



If m = 1, we may take $ = ^i,^- If m = 2, we take 



= Ojy a 



If m > 2, we prove the proposition by induction. Suppose first that 

 the af* +1 ( = 1, ., m) are not all unity, for example, 



l-ag+i^O. 



The left member belongs to the G-F[$P]. Hence we may write 

 126) a(J+ 1 +ft* f + 1 -l, 



ft being a mark =j= in the G-F[p 2 ']. The group therefore contains 

 a substitution of the form 0^\ By 125) and 126), we have 



Assuming our theorem to be true for m 1 indices, the group contains 

 a substitution S f replacing ^ by 



Hence the product 8 ~ S' Off will replace ^ by ^. 

 Suppose on the contrary that 



If the group contains a substitution $ A replacing ^ by | x + | 2 4 ---- h Sm, 

 the product ^ _ ^ ^ 



F -M, n ^ 2, 22 -L m, a lm &i 



will replace |j by /j. But the group will contain a substitution of 

 the form S l if it contains S 2 :n Oi'^^, which replaces j^ by 



