100 CHAPTER II. 



To the operators B t we make correspond the following substitutions 

 of SA(4, 2): 



86) Si^^lMij -Z?2 ' '-^1,1; Bs^S, JD4~_ZJ2 ) 1, -Z?5<~.My, 



where 8 denotes the Abelian substitution of period two: 



li 



0111 

 1011 

 1101 

 1110 



We readily verify that the relations corresponding to the above 

 generational relations are satisfied in virtue of the correspondences 86). 

 Since SA(4$ 2) has the order 



(2 4 -l)2 3 (2 2 -l)2 = 6!, 

 the isomorphism between SA(4, 2) and Gr\ is holoedric. 



119. In determining the factors of composition of the general 

 and special Abelian groups on 2m indices with coefficients in the 

 GF[p n ], we have been led to a quotient- group , SA(2m, p n )/K, 

 where K^ { I, T] is of order 1 or 2 according as p = 2 or p > 2. 

 Owing to the great importance of simple groups, we will designate 

 this quotient -group as A (2m, p n \ it being a simple group except in 

 the three cases m = 1, p n = 2 ; m = 1, p n = 3 5 m = 2, p n = 2, when 



its factors of composition are 2, 3; 2, 2, 3; 2, -*&., respectively. The 

 order A[2m, p n ] of A (2m, p n ) is 



where a == 1 or 2 according as p = 2 or p > 2. 



Conjugacy of operators of period ta) 1 ) m SA(2m,p n ) and A(2m,p n ). 



120. Theorem. - Within the special Abelian group SA(2m, p n ) 

 any substitution S defined ~by 75) is conjugate tvith a substitution Z 

 which replaces ^ and r^ by the respective functions 



lll + yiiyi -f !? ? ftili + d n tji + ^im-l^m-l + 



i w _i = or ese i m =. 

 The theorem is evident if 1/; = y lt - = /3 lz - = d lt - = (i = 2, . . ., m). 

 In the contrary case, we may suppose that a lm , y im , f} lm , d lm are 

 not all zero, first transforming S by P im where i is a certain one 



1) Taken from the author's article, Quarterly Journal, vol. 32, pp.42 63. 



