By 11.0.4, Dp ~ L; so as DpT = TDp, Dp is contained in a Cartan subgroup ofL. As [Dp, t] =I-1, t induces a field automorphism on L. But then either CDp(t) = 1
or t centralizes DP, contrary to the previous paragraph. This completes the proof
of (2). D
LEMMA 11.3.6. k = n.PROOF. Assume k > n and let M := M/CM(VM) and CT(VM) :=QM. By
11.3.4.3, conclusion (2) of 11.2.l holds, so L ~ SL3(q), VM is the sum of twoconjugates of V, k ~ 2n, and m(M, VM) = 2n. We first observe that the weak
closure hypothesis E.6.1 is satisfied with VM in the role of "V"; in particular LT
normalizes CT(V) n CM(VM) = QM, so M = !M(NM(QM)). As m(M, VM) =
2n > 2, m(M, VM) = 2n = s(G, VM) by Theorem E.6.3.
Suppose first that B := V]i ~ T for some y E G with B =I-1. Then m(B) ~
m2(M) = m 2 (L) = 2n = s(G, VM). On the other hand, BE A2n(T, VM) by E.3.10,
so m(B) = 2n. Now assume further that VM ~ Na(B). Then [VM,B] ~ VMnB, so
by symmetry between Band VM, m(VM/CvM(B)) = 2n. Thus from the structure
of the natural module V for L ~ SL 3 (q), B = R2 and V2 = Cv(B). Now for
v E V - V2, [v, BJ = [V, BJ, so by the symmetry between VM and B, v induces a
root element of MY /CMY (B), and V induces the corresponding root group. Thus
[V, VY]= V 1 , and by symmetry, [V, VY]= V{; then y E Na(V 1 ) ~ Mv by 11.3.4.4,
contrary to [V, VY] = Vj_. Thus we have shown that if [VM, V]i] ~ VM n V]i, then
[VM, V]i] = 1.
We next reproduce the argument establishing 11.2.5: Namely we now have
Hypothesis F.7.6 with NM(QM), H, VM in the roles of "G 1 , G 2 , V"; for exam-
ple M = CM(VM)NM(QM) by a Frattini Argument, so VM E R2(NM(QM)) by
11.0.3.1. If Wo(VM, T) ~ 02 (H), then as in 11.2.5, the parameter bM for the graph
as in Definition F.7.8 is even using F.7.9.4, so 1 =I-[VM, V_li.] ~ VM n V_li. for some
g E G by F.7.11.2, contrary to the previous paragraph.
Thus there is B := V_li. ~ T with Bi 02(H). By E.3.20, k 2 s(G, VM) = 2n,
so ask~ 2n, k = 2n. Therefore 02 (H) =KE C(H) with K/0 2 (K) ~ L 2 (2^2 n) by
11.3.5.
Next NaL(VM)(L) is an extension of GL3(q) x GL 2 (q) by field automorphisms.
Using 11.0.4, we conclude that N i!I(L) is an extension of L by field automorphisms.
Therefore for each U ~ VM with m(VM/U) = 2n, CAutM(VM)(U) is a 2-group. Also
m2(AutM(VM)) ~ 3n..
We claim r(G, VM) > 2n. For assume U ~ VM with m(VM/U) = 2n and
Ca(U) i M. Then 1 =/=-V n U, so U contains a 2-central involution. By the
previous paragraph 02 (CM(U)) ~ CM(VM). Thus 02 ' (CM(U)) i CM(VM) by
E.6.12, so conjugating in L if necessary, V1 ~ U. But then Ca(U) ~ Ca(Vi) :::; M,
so the claim is established.
As r(G, VM) > 2n while m(B) = 2n, VH ~ Ca(CB(VH)) ~ Na(B). By
E.3.6, B E Ak(H, VH), so as K ~ L2(2^2 n), B E Syh(K*) is of rank 2n
and CvH(B) = CvH(b) for all b E B#. Thus by G.1.6, VH/CvH(K) is a
direct sum of s 2 1 copies of the natural module for K*. Since VH normalizes
B, m(AutvH(B)) = ks= 2ns, so as m2(AutM(VM)) ~ 3n by an earlier remark,
we conclude that s = 1 and VH/CvH(K) is the natural module for K*. Now
m(AutvH(B)) = k = m(B/CB(VH)), so as Bis the sum of two isomorphic natural
modules for SL3(q), VH induces R~ on B, m([B, VH]) = 4n, and Vi ~ [B, VH].