12.1. A PRELIMINARY CASE: ELIMINATING Ln(2) ON n Ell n* 789so that Tv E Syh(H), B = 02 (H), and Q E Syh(QB). As B 1. M = !M(LT),
there is no 1 =f. R 0 ::::; Q with Ro ::9 (LT, H). Thus Hypotheses C.5.1 and C.5.2 are
satisfied with LT, Qin the roles of "Mo, R". Further LvTv is maximal in LT, so
LvTv = NLr(B). Then we have the hypotheses of C.5.7, and as ILT: LvTvl =f. 2,
C.5. 7 supplies a contradiction.
This contradiction shows that Lv is not normal in Gv. Thus K := Kv > Lv,so as Lv ::9 Mv, K 1. Mv· As Gv E He, K E He by 1.1.3.1. By C.2.6.2,
02,F(K) ::::; Mv ::::; Na(Lv), so K/02(K) is quasisimple by 1.2.1.4. As Lv ::::; Kand
Tv is nontrivial on the Dynkin diagram of Lv/0 2 (Lv), K is not a xo-block, so Q
normalizes K by C.2.4. Thus we have the hypotheses of C.2.7, so K is described
in C.2.7.3. Thus as Tv is nontrival on the Dynkin diagram of Lv/0 2 (Lv) ~ L 3 (2)or L4(2), and Lv ::9 Mv n K, we conclude that case (h) of C.2.7.3 holds with
KTv/02(KTv) ~ Aut(L5(2)) and LvTvnK is the parabolic subgroup determined by
the middle two nodes; in particular n = 4. Let Zv := 01 (Z(0 2 (KTv))), Y := (Z[f),
and (KTv)+ := KTv/0 2 (KTv)· By C.2.7.2, Y is an FF-module for K+Tt ~Aut(L 5 (2)), so we conclude from Theorem B.5.1.1 that [Y, K] = U EB Ut for t E
Tv - NrJU). By B.2.14, Y = [Y, K] EB CzJK). Thus the parabolic LvTv n K
determined by the middle nodes of K centralizes Zv, whereas from the action of
LT on V, Cv(T) is not centralized by LvTv. This contradiction completes the proof
of 12.1.1. D
From Lemma H.9.1, V has the structure of an orthogonal space preserved by
M, and 03 is the set of nonsingular vectors in that space.
LEMMA 12.1.2. (1) If U ::::; V with Ca(U) 1. M, then U is totally singular.
(2) r(G, V);::: n, so that s(G, V) = m(AutM(V), V) = 2.
PROOF. Part (1) follows from 12.1.1 and the fact that 03 is the set of nonsin-
gular vectors in V. Then (1) implies (2). D
Using the lower bound on the parameter r( G, V) in 12.1.2.2, we can apply the
weak-closure machinery in section E.3 (subsection E.3.3) to establish successively
better lower bounds on the parameter w(G, V). Often results are easier to establish
in the case n = 5; for example, the analogue of 12.1.3 below is not established for
n = 4 until 12.1.7.
LEMMA 12.1.3. Ifn = 5 then Wo := Wo(T, V) centralizes V.
PROOF. Suppose that n = 5 but W 0 1. Cr(V). Then there is A:= V^9 ::::; T
with A =f. 1. Recall M = Na(V) and Af9 = Na(A).
We begin by showing we may choose A with m(A) ;::: 5. Suppose first that
V 1. Na(A); then as r(G, V) ;::: 5 by 12.1.2.2, m(A)·;::: 5 by E.3.4.2. So suppose
instead that V::::; Na(A). Here, interchanging the roles of A and V if necessary, we
may assume that m(A) = m(A/CA(V));::: m(V/Cv(A)); equivalently m(Cv(A));:::
m(CA(V)). Suppose that m(A) < 5. Then by our assumption above, m(Cv(A));:::
m(CA(V)) > 5. Hence Cv(A) is not totally singular and 1 =f. Cv 1 (A), so A::::;
T 1 ::::; L. Then as A centralizes a nonsingular vector v E V, by lemma H.9.1.4,
A::::; Lv ~ L4(2) and V = Cv(Lv) EB W, where Wis the sum of a natural module
and its dual. Now m(A) ;::: m(W/Cw(A)), so that A contains a member 13 of
P(Lv, W) by B.1.4.4. Then B.4.9.2iii determines 13 uniquely as J(Cr(v)), so that
13 = J(Cr 1 (v)) =A. ·In particular A is·the unipotent radical -of the stabilizer in
Lv ~ L 4 (2) of a 2-subspace of the natural module W. Thus m(A) = m(V/Cv(A)),