126 CHAPTER V. 



Corollary I. - - HA(2, p 2n ) == A(2, p n } = LF(2, p n \ 



Corollary II. The group of all binary hyperabelian sub- 



stitutions in the GF[p 2n ~] taken fractionally is the group of all 

 linear fractional substitutions in the GF[p n ~]. 



138. In virtue of the transformation of indices, 

 % EE J^ + gg, i? 2 = QJ pn %! 4- p| 2 , 

 where J and Q are primitive roots of the respective equations 



eP B+1 = l, ^+1_~1, 



we have the following identity 



Hence the hyperabelian group on 2m indices with coefficients in the 

 6r_F[p 2w ] is holoedrically isomorphic with the group on 2m indices 

 in the GF[p 2n ] defined by the invariant 



CHAPTER V. 



THE HYPERORTHOGONAL AND RELATED 

 LINEAR GROUPS. 1 ) 



139. We first investigate the linear homogeneous group in the 

 GF[p n ~\ defined by an absolute invariant of the general type 



where each A is a mark =)= of the Q-F\jp f \. 



If r=pVr 1 , we have in the GF[p n ~\ the identity 



t=l 



Hence a substitution which leaves O r absolutely invariant will at 

 most multiply the function 



by a mark t? which satisfies the equations 



1) Dickson, Mathematische Annalen, vol. 52, pp. 561 581. 



