CHAPTER IV. THE HYPERABELIAN GROUP. 



115 



The quotient -group is generated by the substitutions P f j and is thus 

 holoedrically isomorphic with the symmetric group on m letters. The 

 group f is the direct product of m groups each the special linear 

 homogeneous group in the GF[p n ] on q indices ( 103). The sub- 

 stitutions of the i ih group are given as follows 



X' = X 



.., m; 



The structure of the group G (m, q, p n ) is therefore completely de- 

 termined. 



CHAPTER IV. 



THE HYPERABELIAN GROUP. 



129. The totality of linear homogeneous substitutions in the 

 GF[p 2n ~\ 2 m 



S: ;-==^V^ (i = 1, ..., 2m) 



which leave absolutely invariant the function 



2J 1 52 Z 



forms the hyperabelian group 1 ) H(2m, p 2n ). Its name is derived from 

 the fact that the totality of its substitutions whose coefficients belong 

 to the included field 6rF[p n ] constitutes the Abelian group SA(2m,p n \ 

 which is therefore a subgroup of the hyperabelian group. 

 A general substitution S transforms Y into 



a 



if 



!,... ,2m 



J 1,^=1 



The conditions upon S for the absolute invariance of V are thus 



97) 

 where 



2 



1=1 



t, . . ., 2m) 



= 0, unless j and A; differ by unity, when 



1) Introduced by the author, Proc. Lond. Math. Soc., vol. 31, pp. 30 68. 

 It will hardly be confused with Picard's hyperabelian group of infinite order. 



8* 



