LINEAR GROUP WITH QUADRATIC INVARIANT. 183 



ri=r u + r, r,' 4 =r 14 +r 24 



when expressed in the indices 17 . . ., 5 becomes (mod 3) 



12220 



11210 



11120 



12110 



,00001, 



In every case it follows that Q 1 coincides with 0^(5,^). 



190. Theorem. 1 ) -- If p n = 4Z + 1, O' v (6,p n } is Moedrically iso- 

 morpMc with the simple group HA(4c,p* n ). If p n = 4Z -|- 3, Oi(6,^) n ) 

 has the maximal invariant subgroup { I, C C 2 C 3 C 5 C G } of order 2, the 

 quotient -group being holoedrically isomorphie with HA (4, jp 2n ). In 

 either case, 0^(6 ; p") is of index 2 under 0^(6, p") and is extended to 

 the latter by any substitution Oij not a Qij. 



Consider the group H' of quaternary hyperabelian substitutions 

 in the GrF[p* n '] of determinant unity. It has the order 



The special Abelian group SA(4=,p n ) is a subgroup of H'. Denote 

 their second compound groups by -4 4j2 and J3" 4j2 respectively. By 

 161, ^(4,2 leaves absolutely invariant the functions 



XT vy _yy_i_yy 7 y \ y 



4 J: 12 ^34 J: 13- t 24i J: 14- I 23> " -^12 "t" - 34* 



For an arbitrary mark co =f= in the GF[p 2n ~\, the substitution 

 ra 00 



a*-* 11 



00 co- 1 



^00 co f 



is hyperabelian and of determinant unity. Its second compound is 



V Y 



-L to -L 1 



12? 



F' 



14? 



= co**"" 1 F, 



84? 



F ' - F 



'24 



247 



Taking p > 2, we introduce in place of F 12 , F 34 the new indices 



165) fe 



where J is a mark of the G-F[p 2n ] satisfying the equation 



166) J^-^-l, 



1) Bulletin Amer. Math. Soc., May, 1900; Transactions, July, 1900. 



