CHAPTER 1
Structure and intersection properties of 2-locals
In this chapter we show how the structure theory for SQTK-groups from sec-
tion A.3 of Volume I translates into a description of the 2-local subgroups of a
QTKE-group G. We then use this description to establish the existence of certain
uniqueness subgroups, which are crucial to our analysis. We will concentrate on
C-components of 2-locals, and the two families £,(G, T) and 8(G, T) of subgroups
of G discussed in the Introduction to Volume II.In this chapter, and indeed unless otherwise specified throughout the proof of
the Main Theorem, we adopt the following convention:
NOTATION 1.0.1 (Standard Notation). G is a simple QTKE-group, and T E
Syb(G).Recall from the Introduction to Volume I that a finite group G is a QTKE-group
if
(QT) G is quasi thin,
(K) every proper subgroup of G is a JC-group, and(E) F*(M) = 02 (M) for each maximal 2-local subgroup M of G of odd index.
Also as in the Introductions to Volumes I and II, let M denote the set of maximal
2-local subgroups of G, for X ~ G define
M(X) :={NE M: X ~ N},
and recall that a subgroup U ~ME Mis a uniqueness subgroup if M = !M(U).
(Which means M (U) = { M} in the notation more common in the earlier literature).The members of M are of course uniqueness subgroups, but for our purposes it is
preferable to work with smaller uniqueness subgroups, which have better properties
in various arguments involving amalgams, pushing up, etc. We summarize some
useful properties of uniqueness subgroups in the final section of the chapter.
1.1. The collection ?-le
DEFINITION 1.1.1. Define ?-le = 1-l'G to be the set of subgroups H of G such
that F(H) = 02(H); equivalently CH(0 2 (H)) ~ 02 (H) or 02 (F(H)) = 1.
Using this notation, Hypothesis (E)-namely that G is of even characteristic-
just says
M(T) ~?-le.
The property that H E ?-le has many important consequences which we can exploit
later-notably the existence of 2-reduced internal modules for H, such as in lemma
B.2.14. Thus we want ?-le to be as large as possible, so in this section we establish
several sufficient conditions to ensure that a subgroup is in ?-le.
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