LINEAR GROUP WITH QUADRATIC INVARIANT. 



181 



which is the second compound of the special Abelian substitution 



3 

 

 

 



Hence Q' coincides with 0{(5,jp w ). 



Consider next the case 1 a not -square in the GF[p n ~\. Set 



163) 



- 



, 



where a 2 -f 2 = 1. Then O becomes g + g + g + g + g. Hence 

 6r 6 is holoedrically isomorphic with 1 (5,p n ). Reversing equations 163), 

 we get 





- r 



I4 , 



As in 188, we find that Q l (Q expressed in the indices /) contains 

 every Ql\ and the linear substitution (^^Is) an d consequently also 

 $i',4> ^ ne transformed of the former by the latter. 



Expressing in the indices !/ the following substitution of 6r 6 , 



-r. 



y 



^24 



we get CiCtOr'- This 2)4 is not a ^,1 since 2a; 2 -l 

 requires X 2 = /3 2 . But C f 1 C f 4 =?;i belongs to ft- Since 

 not belong to Q ( 164), it foUows that Ol\l 



does 



extends Qi to O t (5, p n ). If O?,'? denotes 0?,'? when expressed in the 

 indices Y t -y, we find that the product K0l\ a $ has the form 



