LINEAR GROUP WITH QUADRATIC INVARIANT. 



179 



12 



13 



2-1 



-34 



having determinant unity. V is seen to be the second compound of 



-1 -1 I 

 -1 -1 -1 

 -1-100 

 ^1-1 



of determinant + 1. Hence ! contains F 126 whenp w =3. 



Since 0' contains the group O.!_i(6,p n ) hut is of index 2 under 

 0_i(6,p), it follows from 182 that O'=01i(6,p"). Applying 

 163, we have the theorem: 



For p n = 4:1 + 3, the group 0_!_i(6,jp w ) is holoedrically isomorphic 

 with the simple group LF^p"') and is of index two under the second 

 orthogonal group 0_i(6,# n ), being extended to it % CgCg. 



189. Theorem. The subgroup 0{(5,# w ) is of index two under 

 1 (5, p n ) and is holoedrically isomorphic with the simple Abelian 

 group A(,p n ). 



By 161, A(4:,p n ), p > 2 is holoedrically isomorphic with the 

 second compound A^ of the quaternary special Ahelian group. 

 ^4,2 leaves absolutely invariant the Pfaffian [1234] and !F 18 + 3^ 4 . 

 By the introduction of the new indices ( 162) 



A^ takes a form not involving Z and so becomes a quinary group Q 

 leaving absolutely invariant the quadratic function 



The group G of all quinary linear substitutions of determinant unity 

 which leave O absolutely invariant will be proven holoedrically iso- 

 morphic with O^S,^ 71 ) and therefore ( 172) of order 



12* 



