586 3. DETERMINING THE CASES FOR LE .Cj(G, T)In this section, in Theorem 3.3.1 we establish an important property of maximal
2-locals containing T and suitable uniqueness subgroups. Theorem 3.3.1 will be
used repeatedly in our analysis of the cases arising from the Fundamental Setup.It turns out that case (2) of Theorem 3.3.1 is not actually required to prove the
Main Theorem, contary to what we expected when we proved the result. However
as the proof for this case is short, we have retained its statement and proof here.THEOREM 3.3.1. Assume G is a simple QTKE-group, T E Syb(G), M E
M (T), and either
(1) LE C*(G, T) with L/0 2 (L) quasisimple and L:::; M, or
(2) XE B*(G,T) with X:::; M.Then Na(T):::; M.
We first record an elementary but important consequence of Theorems 2.1.1 and
3.3.1, that we will use repeatedly in the remainder of the paper: In the Fundamental
Setup, the members of H*(T, M) are minimal parabolics.
COROLLARY 3.3.2. Assume G is a simple QTKE-group, T E Syl 2 (G), and
LE C*(G,T) with L/0 2 (L) quasisimple. Set M := Na((LT)). Then
(1) M= !M((L,T)).
(2) IM(T)I > 1, so H*(T,M) # 0.(3) Na(T):::; M.
(4) For each H E H*(T, M), H n M is the unique maximal subgroup of H
containing T, and HE Ua(T) so that His a minimal parabolic described in B.6.8,
and in E.2.2 when H is nonsolvable.PROOF. Part (1) follows from 1.2.7. Part (2) holds since 2-locals of odd index
in the groups Gin the conclusion of Theorem 2.1.1 are solvable, so that £(G, T)
is empty. Part (3) follows from Theorem 3.3.1. Finally (4) follows from (3) and
3.1.3. []
REMARK 3.3.3. In the simple QTKE-groups G, Na(T) :::; M under the hy-
potheses of Theorem 3.3.1. However there is an almost simple shadow where thisassertion fails: In the extension G of Dl(2) by a graph automorphism of order 3,
there is a maximal parabolic L of E(G) which is an A 8 -block and is a member ofC*(G, T), but which is not invariant under an element of order 3 in Na(T) inducing
the triality outer automorphism on E(G). This extension is of even characteris-tic, but it is neither simple nor quasithin. However it is difficult to verify these
global properties just from the point of view of the 2-local L, so that the shadow
of this group causes difficulties in the proof of 3.3.21.f. Also the proof of 3.3.24 is
complicated by the shadow of the non-maximal parabolic L 3 (2)/2^3 +6 in this same
extension G.Case (2) of the hypothesis of Theorem 3.3.1 will be eliminated fairly early inthe argument in 3.3.10.3. Thus the bulk of the proof is devoted to case (1) of the
hypothesis.NOTATION 3.3.4. In case (1) of the hypothesis of Theorem 3.3.1, where L E
L*(G, T) with L/0 2 (L) quasisimple, set M+ := Lo := (LT). In case (2) of that
hypothesis, where X E B*(G, T), set M+ := X. As Na(T) is 2-closed and hence
solvable, Na(T) =TD, where Dis a Hall 2'-subgroup of Na(T).