See the Introductions to Volumes I and II for terminology used in this overview.
In this first Part, we obtain a solution to the First Main Problem: that is, we
show that a simple QTKE-group G (with Sylow 2-subgroup T) does not have the
local structure of the arbitrary nonsolvable SQTK-group Q with F*(Q) = 02(Q),but instead has more restricted 2-locals resembling those in examples and shadows.
More precisely, we establish the existence of a "large" member of H(T) (i.e., a
uniqueness subgroup of G) resembling a maximal 2-local in an example or shadow.
Then the cases corresponding to the possible uniqueness subgroups will be treated
in subsequent Parts of this Volume.Here is an outline of Part 1:
In chapter 1 we use the results in sections A.2 and A.3 of Volume I to establish
tools for working in 2-local subgroups Hof G, using the fact that our 2-locals arestrongly quasithin. In particular we obtain a good description of the last term
H^00 of the derived series for H, primarily in terms of the C-components of H,
and some information about F(H/0 2 (H)). We then go on to show that certain
subgroups of Gare "uniqueness subgroups" contained in a unique maximal 2-local
M. In particular, we show that members of the sets C(G,T) and S(G,T) are
uniqueness subgroups.
The "disconnected" case where T itself is a uniqueness subgroup and so con-
tained in a unique maximal 2-local, is treated in chapter 2, which characterizes
certain small groups via this property. Consequently after Theorem 2.1.1 is proved,we are able to assume during the remainder of the proof of the Main Theorem that
Tis contained in at least two maximal 2-locals of G .. Hence there exist 2-locals H
with T :::; H f:. M.
Next in chapter 3, we begin by proving two important preliminary results:
Theorem 3.3.1 which says that Na(T):::; M when M = !M(L) with Lin C(G, T)
or 3(G, T); and Theorem 3.1.1, which among other things is needed to apply
Stellmacher's qrc-lemma D.1.5 to the amalgam defined by M and H. The qrc-
lemma gives strong restrictions on certain internal modules U for M via the bound
· {j(AutM(U), U) :::; 2. Section 3.2 then uses those restrictions to determine the
list of possibilities for L/0 2 (£) with LE Cj(G,T), and for the internal modules
V E R2( (L, T} ). This provides the Main Case Division for the proof of the Main
Theorem. One consequence of Theorem 3.3.1 is that members of H*(T,M) are
minimal parabolics, in the sense of the Introduction to Volume II.
The first Part concludes with chapter 4, which uses the methods of pushing
up from chapter C of Volume I to establish some important technical results: In
particular, we show in Theorem 4.2.13 that unless V is an FF-module and L is
"small", then for each I:::; L with 02 (!) f= 1 and L = 02 (L)I, we have M = !M(I).
This large family of uniqueness subgroups then allows us (in Theorems 4.4.3 and
4.4.14) to control the normalizers of nontrivial subgroups of odd order centralizingV. This control is in turn important later, particularly in Part 2 and in chapter
11, when we deal with cases where L/0 2 (£) (or H/0 2 (H) for HE H*(T,M)) is of
Lie type over F2n for some n > 1, allowing us to exploit the existence of nontrivial
Cartan subgroups.