214 CHAPTER VHI. 



a) Let /3 U = 0, fa = d ly = (j = 2, . . ., w - 1), and, if A = 0, 

 also /3 im =d lm =0. If A = A', then must /3 lm =d lm =0 by 214). 

 Evidently S = T^ a Si, where 8 leaves | A and ^ t fixed. By the 

 Abelian relations 78), S 1 involves only the indices | f , ^ (* = 2, ...,w). 

 If $! be not commutative with every substitution Z x of Jj which 

 does not involve | 1? ^ 1; then K will contain a product 



which leaves ^ and ^ fixed. In the contrary case, S is commu- 

 tative with H^^ y . and 3) 2? x; so that, as shown above, S 1 will replace 

 | 2 by ^)| 2 . Since ^ is to be commutative with -M" 2 Jf 3 also, it will 

 replace 7y 2 by ^>^ 2 . Hence, by an Abelian relation, p 2 =l; whence 

 Q = 1. Transforming S by P 12 , we obtain a substitution =f= /which 

 leaves | x and ^i fixed and belongs to K. 



b) Let /3 n = 0, fa, dij (j = 2, . . ., m) be not all zero if I = 0, 

 but let p n = 0, fa, 8ij ( j = 2, . . ., m 1) be not all zero if A = A'. 

 Then by 202, Ji contains a substitution T, affecting only 



which replaces 2 by 



Hence ^T contains $ 17 the transformed of 5 by T. $ x replaces ^ by 

 ajJ! and % by a-^j+fe- 



If J^ contains a substitution F, leaving | 1; %, | 2 fixed, which 

 is not commutative with jfiT will contain 



which leaves i x and ^ fixed. 



In the contrary case, S 1 will be commutative with J5 2 , 3, ^ and 

 Rfli, 3, A and MsM m . Equating the two functions by which S^H^^ X 

 and I?2, s, x$t replace 7; 2 , we find ?g = ( )i 3 +( )i 2 - Equating the 

 two functions by which S\R m ,$, y . and JB m ,s,xi replace ^ TO , we find 

 that Sg = ( )I 3 + ( )im- Hence |g = ^| 3 . Since ^ is to be commu- 

 tative with MS M m , tf 3 = Q%- Then Q = 1. Transforming $,_ by P 13 , 

 we have a substitution =%= I in K which leaves ^ and ^ fixed. 



c) Let f} n 4= 0. We can determine a substitution 5' of K of 

 form similar to that of S but having also d l2 =}= 0. In fact, if A = 0, 

 the products pijdij (j = 2, . . ., m) are not all zero by 214). Trans- 

 forming by a suitable P 2t -, we have ft 2 d 12 =j=0. If A = ', the same 

 result follows unless fiij= dij= (j = 2, . . ., m 1), in which case 

 either /Sun^O or tfim^O by 214). In the latter case, we can take 



