746 io. THE CASE LE .Cj(G,T) NOT NORMAL IN M.and [Z, K] # 1, also I= [I, J(T)] and 1 # U := [Z, J] E R2(I) using B.2.14. Thus
J(T) 1. CT(U) and U is an FF-module. We conclude from intersecting the lists of
A.3.12 and B.4.2 that one of the following holds:
(a) K =I.
(b) I/02(I) ~ SL3(2m), Sp4(2m), or G2(2m).
(c) m = 2 and I/02(I) ~ A1 or A1, with I/C1(U) ~ A1.
(d) I/0 2 (1) is not quasisimple, I= Oz,F(I)K, and Oz,F(J) centralizes U.
By 1.2.1.3, Bo = oz(Bo) normalizes I, so we may assume that K < I, and
so one of (b)-(d) holds. Hence T acts on I by 1.2.1.3, and then also T acts on
K by 1.2.8. In case (d), as CD(Z) = 1, Bon Oz,F(J) :S: T, so Bo acts on the
unique (T n !)-invariant supplement K to Oz,F(J) in J. Suppose case (c) holds.
By A.3.18, I= 03 ' (Mi), so D = D 1 x De, where De := 03 (D) = CD(I/Oz(I))
and D1 := 03 (D) = D n J. As De acts on K, we may assume D1 1. K. Then
D1 E Syh(I). But from the structure of the FF-modules for A1, CD 1 (Z) # 1,contradicting CD(Z) = 1. Suppose case (b) holds. Then K = P^00 for some T-
invariant parabolic Pin I/Oz(I), so as B 0 = 02 (B 0 ) permutes with T, it must also
act on K, completing the proof of the claim.
By the claim Bo acts on K, and by symmetry Bo also acts on Kt if there is
t ET - NT(K). Thus Bo acts on oz(H). Recall that by construction BH acts onD, so D acts on oz(H) n DBH = BH· Therefore [BH, D] :S: BH n D :S: CD(BH)
since the Hall subgroup BH of oz(H n M) is abelian. Now if F # 1, then F does
not centralize [F,D]; thus F = 1, and hence BH = B+ :S: D. Since BHnK is cyclic
of order 2m - 1, while D ~ Z§,,,_i, m divides n.
In the remainder of the proof, we will show that B H = D, and that K # Kt
for some t ET-Ti. Then we will see that the embeddings of Din LLt and KKt
are incompatible.
As M = !M(LoT), Cz( (Lo, H)) = 1. As VK is the natural module for
K/02(K) ~ Lz(2m), Cz(H) = Cz(b) for each b E B=lj;._. Similarly Cz(L 0 ) = Cz(d)
for each d ED#, so as 1 # BH :S: D, we conclude Cz(Lo) = Cz(H) = i. Thus Vi
and VK are natural modules, with CVi(L) = 1 = CvK(K).
Next C.l.26 says that there are nontrivial characteristic subgroups Ci(T) :S: Z
of T and Cz(T) of S, such that one of the following holds: K is a block, Ci(T) :S:
Z(H), or C2(T) :::;) H. As Cz(H) = 1, C 1 (T) 1. Z(H), so either K is a block or H
normalizes Cz(T). Similarly either Lis a block or Cz(T) :::;) L 0 T. However Cz(T)
cannot be normal in both Hand LoT, since M = !M(L 0 T); therefore either Kor
Lis a block.
Next set E := Oi(Z(J(T))) and E 0 := (EL^0 ). By 10.1.2.6 we may apply
E.2.3.2 to LoT, to conclude that Ea = CE 0 (L 0 )V. Therefore as Cz(Lo) = 1,
Ea = V = Vi x V2 is of rank 4n. In particular, E :S: V and E = Ei x Ez with
Ei := En Vi of rank n. Also D =Di x Dz, where Di := D n Li = CD(E 3 -i)·
Notice that CE(d) = 1 ford ED - (D1 U Dz). Similarly applying E.2.3.2 to Hand
using Cz(H) = 1, we conclude that E =EK x CE(K) =EK x CE(BK), where
EK := En VK has rank m = n(H), and BK := B n K. We saw m divides n, so
m :S: n, and hence
So as CE(d) = 1 for each d ED - (D1 U D2), we conclude (interchanging the roles