0.3. AN OUTLINE OF THE PROOF OF THE MAIN THEOREM 489In the next subsection 0.3.2, we describe how to obtain a uniqueness subgroup
U with useful properties, while subsection 0.3.3 discusses how to determine a list of
possiblities for U. Here is a brief summary: No nontrivial subgroup T 0 of T can benormal in both U and H; in particular, Z := D 1 (Z(T)) is not in the center of Y for
some YE {U, H}. This places strong restrictions on the F 2 -module (zY), and on
the action of Y on this module. Our approach concentrates on the situation where
Y is the uniqueness group U. Roughly speaking, we can classify the possibilities
for U and (zu), resulting in a list of cases to be analyzed when Y = U. The bulk
of the proof of the Main Theorem then involves the treatment of these cases, a
process which is outlined in the final subsection 0.3.4.0.3.2. Finding a uniqueness subgroup. We put aside for a while the groups
M and H from the previous subsection, to see how the hypothesis that G is a
QTKE-group gives strong restrictions on the structure of 2-local subgroups of G.We begin with the definition of objects similar to components: For H:::; G, let
C(H) be the set of subgroups L of H minimal subject to
1 =/:= L = L^00 :SJ :SJ H.
We call the members of C(H) the C-components of H. To illustrate and motivate
this definition, consider the following
Example. Suppose G is a group of Lie type over a field F 2 n with n > 1,
and H is a maximal parabolic. If H corresponds to an end node of the Dynkindiagram b.. of G, then H^00 will be the unique member of C(H). But suppose
instead that G is of Lie rank at least 3 and H corresponds to an interior node o
of b... Then the minimality of a C-component L of H says that L covers only that
part of the Levi complement corresponding to just one connected component ofb.. - { o}. Furthermore H^00 is then the product of the C-components of H, and
distinct C-components commute modulo 02(H).
We list some facts about C-components and indicate where these facts canbe found; see also section 0.5 of the Introduction to Volume I. In section A.3 we
develop a theory of C-components in SQTK-groups. Then in 1.2.1 we use this theoryto show that two of the properties in the Example in fact hold for each H E H
in a QTKE-group G: namely (C(H)) = H^00 , and for distinct Li, L2 E C(H),
[L 1 ,L 2 ]:::; 02 (L1) n 02 (L2)·:::; 02(H). The quasithin hypothesis further restricts
the number of factors and the structure of the factors in such commuting products:
If L E C(H), then either L :SJ H, or JLHJ = 2 and L/02(L) ~ L2(2n), Sz(2n),
L 2 (p) with p an odd prime, or J 1. In particular for S E Syl2(H), (Ls) :SJ H,
and (Ls) is Lor LL8 for some s E S. Moreover 1.2.1.4 shows that almost always
L/0 2 (L) is quasisimple. Since the cases where L/0 2 (L) is not quasisimple causelittle difficulty, it is probably best for the expository purposes of this Introduction
to ignore the non-quasisimple cases.
To get some control over how 2-locals intersect, and in particular to produce
uniqueness subgroups, we also wish to see how C-components of H E H embed in
other members of H. To do so, we keep appropriate 2-subgroups S of H in the