LINEAR GROUP WITH QUADRATIC INVARIANT. 191 



group the case m = 4 is shown in 195 196 to be quite except- 

 ional. We denote by F0(m,p n ) the first orthogonal subgroup 

 Oitphftyt when m is odd, and the quotient -group of 0[(m,p n ) by 

 its maximal self - conjugate subgroup {/, (7), when m is even and > 4. 

 By 72, F0(m,p n ) has the order 



F0[m,p n ] = ^(p n ( m -V-T)p n ( m -V(p n ( m -y 

 for m odd; while, for m even, m > 4, 



The second orthogonal group on an even number m > 4 of indices 

 has a simple subgroup 1 ) $0(w,jp w ), previously denoted by O' v (m,p n ) 

 of order 



It wiU be shown in 197198 that this result holds true for m = 4 2 ). 

 In both places, equals 1 according to the form 41 1 of p n . 



Theorem. The first orthogonal group 1 (m,p n ') has for m even 

 and > 4 the factors of composition 2, FO [m, p n ~\, 2 and for m odd the 

 factors of composition 2, F0\m,p n ~\, the case m = 3, p n =3 being 

 exceptional. The second orthogonal group O v (m,p n ) on an even number 

 m>2 of indices has the factors of composition 2, S0[m,p n ]. The 

 ortlwgonal groups on 2 indices are commutative groups. 



195. In virtue of the identity 



u + g + + e - e+i - e+ ----- $ 



it follows from 169 that the group 3 ) L 8 , p n of 2s-ary linear homo- 

 geneous substitutions of determinant unity in the GF[p n ~], p > 2, 



a 



which leave ^ X f Yi invariant is holoedrically isomorphic with 1 (25, J p' 1 ) 



if 1 be a square in the (rJ^lj) 71 ], jp > 2, or if 1 be a not- square 

 while s is even, but is isomorphic with O v (2s,p n ) if 1 be a not- 

 square while s is odd. In particular, L% >p n is, for p > 2, holoedrically 

 isomorphic with 1 (4,^) n ). In determining the structure of L^ p n we 

 do not exclude the case p = 2. 



1) In view of the not -square factor in its invariant, it first appeared in 

 the literature with the notation NS(m, p}. 



2) This result is readily verified for the case pn = 3 not treated in 197198. 



3) The structure of this group was first determined by the author without 

 making use of its isomorphism with orthogonal groups, Proc. Lond. Math. Soc., 

 vol. 30, pp. 7098. 



