624 4. PUSHING UP IN QTKE-GROUPS
particular, Mo is 2-closed, and a Hall 2'-subgroup B of Mo is abelian of p-rank at
most 2 for each odd prime p.
In proving (1), we may take A =I-1. Then 1 ::::; mv(A) ::::; mp(B) ::::; 2 for each
p E n(A). It will suffice to show NH (A) i. M'H, since then as MH is a maximal
subgroup of H, H = (MH, NH(A)), so that (1) holds.
Suppose first that mv(A) = mv(B) for some p. Then A= D1(0v(B)) and so
NH(B) ::::; NH(A). But as B is a Cartan subgroup of K 0 , NK(j(B) i. M 0 , and
this suffices as we just observed.
So assume mv(B) = 2 and mp(A) = 1. Then by E.2.2, one of the following
holds:
(i) K <Ko and K* ~ L2(2n) or Sz(2n).
(ii) K* ~ Sp4(2n).
(iii) K* ~ (S)L 3 (2n).In cases (i) and (ii), there is an element in K 0 - M 0 inverting B, so N K() (A) i. M 0 ,
which suffices to establish (1) in this case as we indicated. Thus we may assume case
(iii) holds, so some t E S acts nontrivially on the Dynkin diagram of K*, and by a
Frattini Argument we may take t E N 8 (B). Then as AS= SA, A is^1 t-invariant. Let
U :=NH (B), [! := U / B, and W the image of NK (B) in U. Then W ~ S3 is
the Weyl group of K and i = sw, where w. is an involution in W, ands E Cu-(W).
Pick preimages w ands of wands. As W acts indecomposably on D1(0v(B)),
s inverts or centralizes B, sos and t act on A, and hence w E NH(A) - MH
completing the proof of (1).
So we may assume the hypotheses of (2). Let D := CB(M+/0 2 (M+)) and Q :=
02(BS). Then, as in the proof of 4.4.4, a Frattini Argument gives S = QNs(B).
Now as M+ :::l M, Ns(B) acts on M+ and hence also on D = CB(M+/02(M+)·
Therefore DNs(B) is a subgroup of G acting on Q, and hence DNs(B)Q = DS is
a subgroup of G, completing the proof of (2). D
Usually we use Theorem 4.4.3 via an appeal to the following corollary:THEOREM 4.4.14. Assume Hypothesis 4.2.1, and letM+ :=(LT), V 0 E R 2 (M+T),
and H E 7-{* (T, M). Assume
(a) V := [Vo, M+] =I- 1, Vo = (Cv 0 (T)M+), and V is not an FF-module for
M+T/CM+T(V).
(b) n(H) > 1.
Then one of the following holds:
(1) 02 (H) n M is 2-closed, and a Hall 2' -subgroup of H n M is faithful on
M+/02(M+)·
(2) M+/0 2 (M+) ~ L 2 (2^2 n), and Vis the n4(2n)-module.
(3) M+/02(M+) ~ L3(2), and V is the core of a 7-dimensional permutation
module for M+/02(M+)·
PROOF. Let Z := 01 (Z(T)) and K := 02 (H). We observed in Remark 3.2.4
that Hypothesis 4.2.1 allows us to apply Theorem 3.1.8. As Vis not an FF-module,
J(T) ::::; CT(V) by B.2.7, so H ::::; Ca(Z), by 3.1.8.3. Similarly by 3:3.2.4, H is a
minimal parabolic described in E.2.2. Since n(H) > 1 by hypothesis, E.2.2 shows
that K / 02 ( K) is of Lie type in characteristic 2 and of Lie rank at most 2, and
Kn Mis a Borel subgroup of K, so in particular Kn Mis 2-closed. Let BH be a
Hall 2'-subgroup of H n M; thus BH is ~belian of odd order.