CHAPTER 3
Determining the cases for L E £j ( G, T)
By Theorem 2.1.1, we may assume in the remainder of the proof of our MainTheorem that the Sylow 2-subgroup T of our QTKE-group G is contained in at least
two distinct maximal 2-local subgroups. Thus we may implement the Thompson
amalgam strategy described in the outline in the Introduction to Volume II: We
choose ME M(T) to contain a uniqueness subgroup of the sort considered in 1.4.1,
and choose a 2-local subgroup H not contained in M. Indeed we may choose H
minimal subject to this constraint:DEFINITION 3.0.1. H*(T, M) denotes the members of H(T) which are minimal
subject to not being contained in M.
In this chapter, we establish two important technical results, and define and
begin to analyze the Fundamental Setup, which will occupy us for most of the proof
of the Main Theorem.We begin in section 3:1 by proving Theorem 3.1.1 and various corollaries of that
result. Theorem 3.1.1 ensures that suitable pairs of subgroups are contained in a
common 2-local subgroup of G. We appeal to this theorem and its corollaries manytimes during the proof of the Main Theorem, but most particularly in applying
Stellmacher's qrc-lemma D.1.5, and in proving the main result of section 3.3.In section 3.2 we define the Fundamental Setup and use the qrc-lemma to
determine the cases that can arise there. A discussion of this important part of the
proof can be found in the introduction to section 3.2.Finally in section 3.3, we prove that if L is in £, ( G, T) or B ( G, T) with M :=
!M( (L, T)) as in 1.4.1, then Na(T) :::; M. We use this result often, most frequently
via its important consequence that each H E H* (T, M) is a minimal parabolic in
the sense of Definition B.6.1.3.1. Common normal subgroups, and the qrc-lemma for QTKE-groups
In this section we assume G is a simple QTKE-group, T E Syl2(G), Z :=
D 1 (Z(T)), and M E M(T). We derive various consequences for QTKE-groups
from Theorem C.5.8 of Volume I, in one case by applying the result in conjunction
with the qrc-lemma D.1.5. We begin with a restatement of Theorem C.5.8.
THEOREM 3.1.1. Assume that M 0 , H E H(T), T is in a unique maximal sub-
group of H, and 1 -=f. R:::; T with RE Syb(0^2 (H)R) and R '.':] Mo. Then there is
1 -=f. Ro:::; R with Ro '.':] (Mo, H).
PROOF. We verify the hypotheses of Theorem C.5.8, most particularly Hypoth-
esis C.5.1: As HE H(T), F*(H) = 02(H) by 1.1.4.6, and as G is a QT KE-group,
m 3 (H) :::; 2. By the hypotheses of Theorem 3.1.1, Tis in a unique maximal sub-group of H-completing the verification of C.5.1.1. Again by those hypotheses,
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