572 3. DETERMINING THE CASES FOR LE .C'j (G, T)R :SJ Mo and R E Syb(0^2 (H)R), so C.5.1.2 holds. Thus Hypothesis C.5.1 is
indeed satisfied, while by the hypotheses of this section, T E Syl2 ( G) and G is a
simple QTKE-group, supplying the remaining hypotheses of Theorem C.5.8. Of
course the conclusion of C.5.8 is the existence of a nontrivial normal subgroup of
(M 0 , H) contained in R, so Theorem 3.1.1 is established. D
We sometimes use the following easy observation:
LEMMA 3.1.2. If T:::; Y:::; HE 1-l(T), then also YE 1-l(T) ~ 1-le.PROOF. As HE 1-l, 02(H) # 1. Further TE Syb(Y), so 1 # 02(H) :::; 02(Y)
by A.1.6, and hence also Y E 1-l. Finally Y E 1-le by 1.1.4.6. DIn view of Theorem 2.1.1, we may assume that our fixed ME M(T) is not the
unique maximal 2-local subgroup of G containing T, so that 1-l*(T, M) is nonempty.
During the remainder of our proof of our Main Theorem, we typically implement
the Thompson amalgam strategy exploiting the interaction of M with some member
of 1-l*(T, M).
Recall also from Definition B.6.2 that a subgroup X of G is in Uc(T) if Tis
contained in a unique maximal subgroup of X; and X is in Uc(T) if X E Uc(T)
and Tis not normal in X. In the terminology of Definition B.6.1, the members of
Uc(T) are called minimal parabolics.
As mentioned in the Introduction to Volume II and at the start of this chapter,
once we have established Theorem 3.3.1 in the final section of this chapter, part (2)of the next lemma will ensure that members of 1{* (T, M) are minimal parabolics
for suitable choices of M.
LEMMA 3.1.3. Assume HE 1-l*(T,M). Then
(1) H n M is the unique maximal subgroup of H containing T. That is,1-l*(T,M) ~ Uc(T).
(2) If Nc(T) :::; Mor His not 2-closed, then HE Uc(T). Thus His a minimal
parabolic, and so is described in B.6.8, and in E.2.2 if H is nonsolvable.PROOF. Since H i. M, T :::; H n M < H. If T :::; Y < H, then by 3.1.2,
YE 1-l(T); thus Y:::; H n M by the minimality of Hin the definition of 1-l*(T, M),
so that (1) holds. If Nc(T) :::; M or His riot 2-closed, then Tis not normal in H,
so (2) holds. DLEMMA 3.1.4. Assume that H:::; G and V is an elementary abelian 2-subgroup
of H n M such that V is a TI-set under M with Nc(V) :::; M and H:::; Nc(U) for
some 1 < U:::; V. Then
(1) H n M = NH(V).
(2) Hi.Miff Hi. Nc(V), in which case H n M = NH(V) < H.
PROOF. As we assume Nc(V) :::; M, NH(V) :::; H n M. Conversely as Vis aTI-set iri M, NM(U) :::; NM(V). Then as H :::; Nc(U) by hypothesis, H n M =
H n NM(U):::; NH(V), so that (1) holds. Then (2) follows. D
Usually we will apply Theorem 3.1.l under one of the hypotheses in Hypothesis