LINEAR HOMOGENEOUS GROUP IN THE 



etc. 



209 



207. Theorem. The senary second hyperdbelian group Jv in 

 the 6rJF[2 n ] is a simple group holoedrically isomorphic with HA(4,2 2n ). 



We begin as in 190, but make the following transformation 

 of indices, including the transformation 204) for m = 3: 



il ^14> fe = Y IS , % = ^23> ^2 = ^24? 



6,=A/tf-ir 18 + 0A-v*r 34 , % = ^/ 2 (r 12 + r 34 ). 



The invariant of the second compound group is transformed thus: 



r 12 r 34 - r 13 r 24 + 



If we take 



the substitution 206) becomes in the new indices a substitution 202) 

 with coefficients in the 6rF[2 n ]. In particular, if to be a suitable 

 primitive root of the 6rF[2 2n ], r will be the primitive root g of 

 ^ 2W + 1 = 1. We thus reach, by 207), the substitution L. 



We next express in the new indices the general substitution [a] 2 , 

 given in 164, of the second compound At, 2 of the group of qua- 

 ternary Abelian substitutions of determinant unity in the GF[2 n ~\. 

 For example, it will replace | 2 = Y 13 by 



by one of the Abelian conditions, while A 2 ^- 1 -^ <7 -f 1 = by 203). 

 Proceeding in this manner, we find that [a] 2 takes the form 



