LINEAR GROUP WITH QUADRATIC INVARIANT. 193 



The same result holds if the substitutions be compounded in reverse 

 order, so that the substitutions are commutative. Further, the only 

 substitutions belonging to both of the sets 181) and 182) are seen to be 



184) 8{ = 6i, ^ = <^1, %2 = <*%2, *7 2 = %- 



The substitution 181) leaves J^ -j- | 2 ^ 2 absolutely invariant if 

 and only if ad 0y = 1. Hence there are (p 2n T)p n such substitu- 

 tions. It follows that there are 



y {(!" -I)!*}' (ifi2) 



distinct substitutions 183) for which 



185) ad-/ty=l, AD-BC = 1. 



The substitution T 2)X , denned in 114, wiU be of the form 183) 

 only if 



Therefore A= a" 1 , D = KCC~\ d - x- 1 *, so that 



ad $y = x- l a 2 AD - BC = xa-*' 



It will thus satisfy the relations 185) only when x is a square 

 in the Gf[p n ]. Hence there are at least {(p 2n T)p n } 2 substitu- 

 tions 183) which satisfy the single relation 



186) (ad - p<y)(AD-BC) = 1. 



For p > 2, L^ p n is holoedrically isomorphic with #i(4,^ n ) and 

 therefore, by 172, has the order (p Sn p n )(p 2n T)p n . Hence 

 L^ p n is composed of the substitutions 183) alone. Those of these 

 substitutions which satisfy 185) form a subgroup Li lP * of index two. 

 It is extended to the main group L^ p n by a substitution T 2jX . 



For p = 2, the substitutions 183) which satisfy 185) form a sub- 

 group jC 2 ,2 of index two under Z 2j2 . In fact, by 204, the order 

 of L^n is 2(2 2 -l) 2 2 2w , which is double the order of L 2 >. The 

 transposition (ii%) serves to extend Z 2?2 to i 2 , 2 ; for, if 183) reduces 

 to the form (| A ^), then a A == aC = aB = 0, ccD = 1, whence 



For either p > 2 or > = 2, the group i^^w of the substitutions 

 183) satisfying 185) has an invariant subgroup formed of the sub- 

 stitutions 181) which satisfy the relation ad fty = 1. The quotient- 

 group is holoedrically isomorphic with the simple group LF(2,p n *). 

 Indeed, it is clearly the quotient -group of the group of substitu- 

 tions 182) satisfying AD BC = 1 by the group of the substitu- 

 tions 184), a 2 =l, common to the two sets 181) and 182) under 

 the conditions 185). 



DlCKSON, Linear Groups. 13 



