V{ Thus Vi is an £ 2 -conjugate of Vi, and hence we may assume g E Na(Vi). In
particular Vf ::::; Vi. Now by 11.1.3, g E Na(L2), so g permutes the fixed points of
Sylow 2-groups of £ 2. Of course V 1 L^2 is the set of these subspaces, so conjugating
in £ 2 , we may assume g E Na(V 1 ), contradicting an earlier observation. Thus
[A, V] < V 2 , so as V 2 =[A, VJ, Cv(L) =/=-1, and hence L ~ Sp4(q).
Now V 2 = Cv(A), so m(Autv(A)) = 2n, and by symmetry m(A) = 2n. If
some a EA induces an F q-transvection on V, then without loss [V3, a] = 1. Hence
% is the root group of a transvection on A, and then by symmetry A contains the
root group centralizing V3 and [V 3 , A] = V1. Then m(A/CA(V3)) = n, contrary to
the claim applied to V3 in the role of W. Therefore A contains no transvections.Let R1 and Rk be root groups of transvections contained in R2, S := R1Rk,
and As= An S. Then m(S/As)::::; m(R2/A) = n, so as R1nA=1, we conclude
S = Rz x As and m(As) = n. Now Rk ::::; Y ::::; C1(R1) with Y ~ L2(q), andsetting Vy := [V, Y], Vy is a nondegenerate 2-dimensional F q-subspace of V with
Vy::::; Cv(R1). Indeed taking Rk :s! f', V1 = [Vy,Rk], so Vi= [Vy, SJ= [Vy,As];
hence as V 2 = [V, A], Vi = [V, A]. This contradicts an earlier reduction, and
completes the proof. D
LEMMA 11.2.5. Let HE H(T) with H 1. M. If L ~ SL3(q), assume furtherCa(Vi) ::::; M. Then either
(1) Wo(T, V) 1. 02(H), or
{2) (VNa(V1)) is nonabelian.PROOF. We observe that Hypothesis F.7.6 is satisfied with LT, Hin the roles
of "G1, G2". Adopt the notation of section F.7, and assume W 0 (T, V) ::::; 02(H).
Then the parameter b of Definition F.7.8 is even by F.7.9.4. Thus by F.7.11.2,
there exists g E G with 1 =/=-[V, Vg] ::::; V n V9, so (vNa(Vi)) is nonabelian in view
of 11.2.4.2, and hence (2) holds. D11.3. Eliminating the shadow of L4(q)
Notice that when L ~ SL3(q), the condition Ca(Vi) ::::; M distinguishes L 4 (q)
from the other shadows. In this section, we eliminate that troublesome configura-
tion, and also (when we show Cv(L) = 1) eliminate the shadow of Sp6(q) in the
case L ~ Sp4(q).
Throughout this section we assume:
HYPOTHESIS 11.3.1. (1) There exists HE H*(T, M) with [Z, HJ =/=-1.
(2) If L ~ SL3(q) then Ca(V1)::::; M. ·
The object of this section is to prove:
PROPOSITION 11.3.2. Assume Hypothesis 11.3.1. Then
{1) L ~ Sp4(q).(2) Ca(V1) 1. M.
(3) Cv(L) = 1.
(4) If Wo(T, V) ::::; 02 (H), then (VCa(Vi)) is nonabelian.
During this section we assume the pair G, L is a counterexample to Proposition
11.3.2. We begin a series of reductions.