16.4. INTEB.SECTIONS OF NG(L) WITH CONJUGATES OF CG(L) 1187Since we saw 02(Lr) ~ L', we conclude by symmetry that 02 (Lr) :::1 M when
Z(L) = 1.
.Suppose that Tc > (z) and set Sc := Nr 0 (V). Then as [V, Sc] = [R, Sc] ~
V n Tc = (z) = Cr 0 (R), Sc is of order 4 and st = (t+), where t+ _induces
a transvection on V with center (z) and axis 02 (Lr)(z). By symmetry there is
to EM n Tf; - R which induces a transvection on V with center (r) and axis not
containing z. As (z) is the center oft+, CM+(t+) ~ CM(z)+ =st x NL(V)+.
In particular S-6 = 02(CM+(t+)), so either 02(M+) = 1 or 02(M+) =St. But
in the latter case, M+ acts on z, whereas [t 0 , z] # 1. Thus 02 (M+) = 1. Let
x+ := (t+, tci); then x+ ~ S3 centralizes 02(Lr) from the structure of subgroups
of the general linear group generated by a pair of transvections.Assume first that Z(L) = 1. Then 02 (Lr) :::1 M and NL(V) centralizes
V/02(Lr), so
[x+,NL(v)+]::::; CM+(02(Lr)) n CM+(V/02(Lr)) ~ 02(M+) = 1
by Coprime Action, and hence NL(V) normalizes [V, X] = (z, r). This is _a con-
tradiction as r does not centralize Lv. Therefore Z(L) # 1, so M is irreducible
on V by earlier remarks. As M+ contains the transvection t+, with CM+ (t+) ~CM(z)+ ~ Z2 x S5, it follows from G.6.4 that M+ ~ S1 and V is the natural
module. This is a contradiction as the noncentral chief factor for NL(v)+ on V
is the natural module for L 2 (4), whereas the centralizer in A1 of a transvection
has the A 5 -module as its noncentral chief factor. This contradiction completes theelimination of the case Tc > (z).
So Tc= (z). Then T = TLTcR acts on V, so T+ = Tt ~ Ds is Sylow in M+.
We may apply A.3.12 to conclude that Lv ~-NE C(M), and the embedding of Ltin N+ is described in A.3.14. As D 8 is Sylow in N+r+ and there is a nontrivial
F 2 N+T+-representation of dimension 6,· we conclude that either N+ = Lt or
N+ ~ A 7 • In the first case, M acts on Cv(Lv) = (z), and then by symmetry,
M also centralizes r, a contradiction as Lv does not centralize r. In the second
case, as m(V) = 6 and S5 ~ NL(V)+ = CM+(z), we conclude that Vis the core ofthe 7-dimensional permutation module for M+, and that z is of weight 2 in that
module. This is impossible, as we saw at the end of the previous paragraph. This
contradiction completes the proof of 16.4. 7. DLEMMA 16.4.8. Let r be an involution of R. Then either
(1) Lr~ L'; or
(2) 031 (H*) ~ PGL3(2n) or L~'^0 (2n), 2n = E mod 3, r induces an inner
automorphism on L, n # 3, and 03 (Lr) ~ L'.
PROOF. As usual recall that Lr~ Cc(r) ~ H'. Let fl':= H'/K', and defineA(Lr) :=(OP' (Lr): pis an odd prime such that mp(L) > 1).
Applying 16.4.5 with L', r in the roles of "L, i", either A(Lr) ~ LB or 031 (H') ~
PGL3(2n) or L~'^0 (2n).Assume first that the latter case holds. In particular, n 2: 2.
We will first treat the subcase where r induces an outer automorphism on L.
Then by 16.1.4, either Lr is PSL3(2nl^2 ), U3(2nl^2 ) with n > 2, or .L2(2n); or r
induces a graph-field automorphism on L* ~ L 3 (4) and Lr-~ Eg. As m3(L') =