102 



/CHAPTER II. 



d 12 = = di m = 0, we have reached X. If di w =f=0, we transform 

 $1 by Q-2,m,a, where d 12 -f <7#i m = 0, and reach a substitution of the 

 form S but having also # 12 = 0. In a similar manner we make 

 tf 18 = . - = di m -.i = and reach I. Finally, if 8 lm = but d 12 , 

 d 13 , . . ., dim i are not all zero, we may suppose that #i m _ i =j= 0, 

 first transforming by some P im i. We then transform it by Q^ m i^j 

 for i = 2, 3, . . ., m 2 in succession, and make 



so that we reach X. 



Corollary. - - If a u , y u , fin, $u (i = 2, . . ., m) are not all zero 

 in Sj it is conjugate within SA(2m, p n ) with one of the two types of 

 substitutions : 



Since the conjugate substitution I! then has i m =f= 0, we may 

 transform it by T mj i m . Then if di m _i=0, we have Z r In the 

 contrary case, we transform also by lm \i dim i and get Z 2 . 



121. Theorem. - - The special Abelian group SA(2m, p n \ p> 2, 

 contains exactly m sets of conjugate substitutions of period 2. I he 

 r th set includes 



substitutions all conjugate with T r = li,_i ^2, i ^V, i- 



In order that the special Abelian substitution 75) shall be 

 identical with its reciprocal 77), for ^ = 1, it is necessary and 

 sufficient that 



Every substitution of period 2 of SA(2m,p n '), p > 2, has therefore 

 the form 



For m = l, we have a^ = 1, so that TI ? _I is the only sub- 

 stitution /S. In order to prove the first part of our theorem by 

 induction, we assume that every special Abelian substitution in the 



