THE ABELIAN LINEAR' GHOT^P. 101 



of the integers 2, 3, . . ., m. According as a\ m 4= 0, y im 4= 0, ft lm ^0 

 or dim 4= 0, we transform S by J, Jf /n , Jf 1? or M\M m respectively 

 and obtain a substitution S' in which a lw =J=0. Transforming S' 

 by Z,,^, we obtain a substitution /S" which replaces ^ by 



r- 



Since i m =)= ^ we can choose I in the field to make the coefficient 

 of tj m vanish. Transforming S" (in which now cc lm =[=0, y lm = 0) 

 by L\,Q 9 we reach a substitution S^ which replaces | 17 % by 

 respectively 



, 



On - Wii)* + yn 1 ?! + + 



( )l+ ( HiH ----- h (ft TO + 0a lm )| OT + 



We choose p to make /?i,4- (>!= 0. Hence S l has i m =J=^ 



yim= 01m = 0. 



We next determine an Abelian substitution which affects only 

 the indices 3? ??2> t>m, ^m and which transforms S^ into a substitution 

 5 2 having lOT ^0, y lm = /3 ir;2 = y 12 == /3 12 = 0. 



a) Let cr 12 = y 12 = 0. If $ 12 = 0, the transformed of S 1 by M. 2 

 gives S 2 . If ft 2 and d\ 2 are both not zero, we transform S { by .L^, 

 where /3 12 Q d 12 = 0, and obtain S 2 . 



b) Let 12 and y 12 be not both zero. Transforming by M 2 

 when y 12 =(= 0, we may suppose that 12 =[= in S v Transforming it 

 by L^Q, we can make y 12 = 0. If then ^ 12 =)= 0, we transform by 

 is, ? and make /3 12 = 0. Suppose, however, that ^ 12 = 0. If di m =)=0, 

 we transform by R^m,^) where /S 12 -f- ^ ^i = 0, and reach /S'g. But 

 if di m =0 ; we have ^ if 12 =0; while for /3 12 4=0, we transform 

 by Q^qMs, where a l2 Qai m = 0, and reach $ 2 . 



In an analogous manner, we can determine an Abelian sub- 

 stitution which affects only | 3 , ?? 3 , i ro , rj m and which transforms ^ 

 into a substitution $ 3 having 



lm 4= ^ y = fe = yi8 = As = rim = ftm = 0. 



Repeating the process, we may also make 



We therefore reach a substitution S conjugate with S within the 

 special Abelian group and replacing | n ^ by respectively 



Transforming 5 by m,2, a, where 12 <?ai TO = 0, we obtain a 

 substitution of the form S but having 13 == 0. Similarly, we may 

 make a 13 = = i m _i = 0. If, in the resulting substitution S 19 



