142 CHAPTER V. 



Let m 5 4. If be not commutative with every 



= Otf (i,j = 3, . . ., m; a* 1 +1 + 

 then I contains the substitutions leaving f^ and | 2 fixed, 



not all of which reduce to the identity. In the contrary case, X must, 

 by 147, have the form 



l! = fi>& (*-3,...,m). 



Hence I contains the product 



which does not reduce to the identity; for, if so, v, = a and S would, 

 contrary to the hypothesis made in 148, have the form 



! = o& (i = 1, 2, . . ., m). 



151. Theorem. Except in the case m = 3, jp* = 2, $ 

 coincides with the group H w ,^ jS . 



The proofs of the theorems of 149 150 hold for any value of 

 m 5 3. Hence by a repeated application of these theorems, we finally' 

 reach in the group I a substitution S =j= 1 leaving m 2 indices 

 fixed and therefore of the form O^J, we may assume. If it reduce 

 to Cj(7 2 , when p =|=2, its transformed by 0"]% gives the substitution 



so that I will contain an Oi )3 neither the identity nor C^Cg. Indeed, 

 by 144, there exist solutions =(= 0, /3 =)= in the 6rF[j? 2 *], p*> 2 ; 

 of the equation a^+ x + /3^ v + 1 = 1. Hence I contains a substitution 

 Oi,'? neither the identity nor C C 2 . It follows then from 144 that, 

 for p s > 3, I contains every substitution O"^. Transforming by sub- 

 stitutions of the form ( ; |,)<7/, we obtain in I every OfJ. 



These substitutions suffice, except when m ^> 3, p* = 2, to 

 generate the group H W , P)S . Indeed, by applying the formula 



where 



{= aP'+i -h r- 1 /3^+S P' = ap(r~ l - 1); T*'* 1 = 1, 



it follows from 146 that every substitution of G- m , p , s has the form 

 h or }iT m ^ where h is generated from the Ofy and has determinant 

 unity. Hence the substitutions of H W5J9) . S (of determinant unity) are 

 of the form h. 



