104 



CHAPTER II. 



Taking a 11 + tf# = and transforming the resulting substitution 

 by T^t, we obtain PI = (Slim) (ififyn)- The latter is transformed 

 into Ji 5 _ i by the following Abelian substitutions (and by no others): 



K ml 



'ml 



L 2(a wil d x m i 



iKi)-! 1 



Wi iymi) = 1 J 



It follows that, if d =f= 0, Zj is conjugate with some T r . 



Case b). d = in Z r The, Abelian conditions 79) now give 



y,-m=tf,-m=0 (i = 2, ..., m-1), an+a TOIB =0, < = !. 

 Transforming Zj by 1, , *, where 1 2>La n = 0, we obtain 



u ... 01 

 u ... 



a . , 



-fc, H 



. - 



Hence TT=Ti i1 TT f , where TT' affects only &, ij.-.(i = 2, . . ., m) 

 and may therefore, by hypothesis, be transformed into a product of 

 the T^ i by an Abelian substitution on the same indices. It will 

 transform W into a product of the T^_! (j = 1, , . ., m), which is 

 conjugate with some T r . 



Case c). In virtue of the Abelian conditions, Z 2 becomes 



Transforming Z 2 by the product Qi, m ,iQm--i,i, a, where 1 2Aa u = 0, 

 1 + 2tfa n = 0, we get a similar substitution but having zeros in place 

 of the four elements 1. Since it is of the form W, we may proceed 

 as in case b). 



