84 CHAPTER I. 



The resulting substitution V~ l T V belongs to J and leaves ^ (i > 2) 

 unaltered: 



If, however, every a,-i = 0, T itself leaves fixed m 1 indices. In 

 either case, J contains a substitution =J= JT of the form 1 ) 



rfi = rii (* = 3, . . ., m). 

 Then J" contains the two substitutions leaving 7^3, . . ., rj m fixed: 



= ^2 



7?-l T>-1 T? T? I 1 ?! = ^1 



= H ^2,3^1-^-^2,3^' \ , 



1^2 = ^2 



These substitutions are both of the form 



If T 2 and T 3 reduce to the identity, R itself becomes 



JRi: 1?1 = ^1 -f ^13^3 H ----- h yimllm, ^2 = ^2+ ^23^3 H ----- h 



If y\j= y%j= (j = 4, . . ., m) 9 this substitution =|= J is of the 

 form C7". In the contrary case, we may suppose that y 14 and y 24 are 

 not both zero. Then 



is a substitution =}= / of the form U and belonging to J. Hence, 

 in every case J contains a substitution U not the identity. For 

 definiteness, let c?i =j= and introduce the new indices 



Then 7 becomes -Bi,3, ff| . Transforming the latter by the substitution 

 61-Ag,, Si = -l- 1 | 2 , |! = fc (t-3, ...,m), 



where A is an arbitrary mark =)= of the 6r_F[j9 n ], we reach in J 

 the substitution BI^IO, and therefore every ^1,3,^. The latter is 

 transformed into Bt,a f i(k =(=1? 3) by the following substitution of f: 



&- -^ a -i,, !-$ (*- 2, :..,*- !,* + !, ..., w). 



1) From this point, the proofs by Burnside and Jordan (1. c.) are incomplete. 

 The specific errors were made in the Traite, p. 108, 1 and in The theory of 

 groups, p. 316, "This process may now be repeated", etc. 



