10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 745(4) Case {1} of 10.1.1 does not hold; that is, n(H) = 1 for each Hin that case.
PROOF. Let EH be a Hall 2'-subgroup of H n M. Notice EH permutes with
T, so that B+ :=EH n Lo permutes with To.
We first establish (2). If Vis not an FF-module for LoT/CLoT(V), then (2)
follows from Theorem 4.4.14; so we may assume that B := CBH (Lo) i= 1 and Visan FF-module for Laf'. We first verify Hypothesis 4.4.1 and then we apply Theorem
4.4.3: By 4.4.13.2 we have BT= TB, giving (1) and (2) of Hypothesis 4.4.1. As
BT= TB, NH(B) 1:. M by 4.4.13.1. As Vi :SI 02 (M), B acts on Vi. But by 10.1.3,([B[, [LI) = 1, so as [Endt/\/i)[ divides [L[, [V, BJ = 1. Thus we also have 4.4.1.3,
with V in the role of "VB". Since L < Lo, case (1) of Theorem 4.4.3 must hold,
contradicting our earlier observation that NH(B) 1:. M. So (2) is established.
Appealing to (2), 10.1.3, and the structure of Aut(Li), we conclude that either
(i) L is not L3(2) and EH = B+F, with B+ :::; Bo (since B+ permutes with
To), and F induces field automorphisms on L 0 , or
(ii) L ~ L3(2) and EH = B+ :::; Lo.Assume first that (ii) holds; this case corresponds to cases (3), (5), and (6)
of 10.1.1. Then as EH permutes with To, EH is a 3-group, and so n(H) = 2.
Further BH02(BHT) is T-invariant, so BHT contains a Sylow 3-group of L 0 , and
hence BHTo = 02 (H n M)To is a maximal parabolic in L 0. In particular, (H n
M)/0 2 (H n M) ~ 83 wr Z 2 , and the only case in E.2.2 with n(H) = 2 satisfying
this condition is H/0 2 (H) ~ 85 wr Z 2. For example case (2b) of E.2.2 is ruled
out as here (H n M)/02,3(H n M) ~ Ds. Thus we have established (3), and alsoproved (1) in this case. So from now on, we may assume that (i) holds.
Suppose next we are in case (2) of 10.1.1, where L ~ A5. Then F = 1, so that
EH= B+:::; B 0. Now we may argue much as in the previous paragraph: As EHpermutes with T, it is a 3-group and so n(H) = 2, completing the proof of (1) and
hence of the lemma in this case.
So at this point, we have reduced to one of cases (1), (4), or (7) of 10.1.1. Since
EH= B+F by (i), there is a EH-invariant Hall 2'-subgroup D of Bo, and B+ :::; D.
By 10.1.2.5, Cn(Z) = 1 in cases (1) and (7) of 10.1.1, while Cn(Z) ~ Z~n+l in
case (4). Further in any case, Cp(Z) = 1.
Suppose first that [Z, H] = 1. Then F = Cp(Z) = 1, so B+ =EH :::; Cn(Z),and hence CB 0 (Z)-/= 1 so that case (4) of 10.1.1 holds by the previous paragraph.
Set m := n(H) ;:=: 2. From E.2.2, EH has a cyclic subgroup B of order 2m - 1. As
B :::; Cn(Z), 2m - 1 divides 2n + 1, so m divides 2n. If m divides n then 2m - 1
divides 2n - 1, impossible as (2n + 1, 2n - 1) = 1. Thus m = 2d is even and d
divides n, so as (2n + 1, 2n - 1) = 1, 2d - 1 = 1 and hence m = 2. Therefore the
lemma holds in this case.We may now assume that [Z, H] -/= 1. Then Lo = [L 0 , J(T)] by 10.2.2.1,
eliminating cases (4) and (7) of 10.1.1, leaving only case (1), where it remains to
derive a contradiction in order to complete the proof of the lemma. Recall in this
case that Cnp(Z) = 1.
By 10.2.2.2, 02 (H) = [0^2 (H), J(T)]. By E.2.3.1, 02 (H) = (KT) where KE
C(H) with K/0 2 (K) ~ L 2 (2m) or A5, and setting W := (ZH) and VK := [W,K],
VK/CvK(K) is the natural module for K/02(K). Observe VK is not the A5-module
as EH:::; DF and Cnp(Z) = 1, whereas ifVK were the A5-modulethen [Z, EH]= 1.