154 



CHAPTER VI. 



For p n = 2 n or p n = 4? -j- 3, 6rl )2 is a simple group lioloedrically 

 isomorpMc ivith LF(4:,p n ). For _p n = 4Z + 1, 6ri )2 has a maximal 

 self -conjugate subgroup [I, T] of order two, the quotient -group being 

 holoedrically isomorphic with LF(4, p n ). If e = 1 or 2 according as 

 p = 2 or p > 2, the order of G[^ is 



164. Theorem. -- The second compound G^% of the general linear 

 homogeneous group G in the 6r.F[j> n ] contains the substitution 



14(Vi V vY Y f Y Y' Y Y' Y 



/ 12 12? 18 13? 14 14? 2S 2 



Y 



- 24 - 1 24? 



ifj and only if, v be a square in the field. 



To the substitution (a,-,-) of Gr 4 corresponds in 

 stitution [a] 2 : 



-*- 1 -> -./- ( O JL { A JL )Q JL & A 



Y 1 

 J -~~ 



Y' 



- L ~ 



the sub- 



-34 



Consider the "partial substitution ", possibly of determinant zero, 



-23 



24 



84 



141) 



Y' 



-J- OQ 



Y' = 



J1 24 



Y' 



- I 34 



Its determinant is readily seen to equal 



23 



24 



C 42 



C 43 



