492 INTRODUCTION TO VOLUME II
(l)02((M 0 ,H)) =I-1, so Mo is not a uniqueness subgroup of G.
(2) Vi. 02(H) and q(Mo/CM 0 (V), V)::::; 2.
(3) q(Mo/CM 0 (V), V)::::; 2.
When we apply this result with Mo our uniqueness subgroup U from subsection
0.3.1, case (1) does not arise, so the module V satisfies ij_::::; 2.
In section D.3, we determine the groups and modules satisfying this strong
restriction (and a suitable minimality assumption) under the SQTK-hypothesis.
Since the most general SQTK-group H of characteristic 2 could have arbitrary
internal modules as sections of 02 (H), Theorem 3.1.6 leads to a solution in section
3.2 of the First Main Problem for QTKE-groups:
First Main Problem. Show that a simple QTKE-group G does not have
the local structure of the general nonsimple strongly quasithin JC-group Q with
F*(Q) = 02 (Q), but instead has a more restrictive structure resembling that of the
examples in the conclusion of the Main Theorem, or the shadows of groups with
similar local structure.
A solution of the First Main Problem amounts to showing that there are
relatively few choices for L/0 2 (L) and its action on V, where L E .Cj(G, T),
V E R 2 ((L,T)), and [V,L] =I- 1. Indeed in most cases, L/02(L) is a group of
Lie type in characteristic 2 and V is a "natural module" for L/02(L). This leads
us in section 3.2 to define the Fundamental Setup FSU (3.2.1), and to the possibili-
ties for L/0 2 (L) and V listed in 3.2.5-3.2.9. The proof can be roughly summarized
as follows: Apply Theorem 3.1.6 to Mo := U = (L, T). As Mo is a uniqueness sub-
group, conclusion (1) of 3.1.6 cannot hold. Then from section D.3, the restrictions
on q and q in conclusions (2) and (3) of 3.1.6 allow us to determine a short list of
possibilities for M 0 /CM 0 (V) and its action on V.
0.3.4. Handling the resulting list of cases. We continue to restrict atten-
tion to the most important case where L E .C*(G, T) with L/02(L) quasisimple,
and let Lo := (LT) and M := Na(L 0 ). Then in the FSU, there is 1 =I- V =
[V, Lo] E R2(LoT) with V/Cv(Lo) an irreducible LoT-module. Set VM := (VM)
and M := M/CM(VM)· By 3.2.2, VM E R2(M), and by Theorems 3.2.5 and 3.2.6,
we may choose V so that one of the following holds:
(1) V = Vivi :sJ M.
(2) Cv(L) = 1, V :sJ T, and Vis a TI-set under M.^5
(3) L ~ L 3 (2), L <Lo, and subcase 3.c.iii of Theorem 3.2.6 holds.Further the choices for L and V are highly restricted, and are listed in Theorems
3.2.5 and 3.2.6, with further information given in 3.2.8 and 3.2.9.
The bulk of the proof of our Main Theorem consists of a treatment of theresulting list of possibilities for L and V. The analysis falls into several broad
categories: The cases with ILT I = 2 are handled comparatively easily in chapter
10; so from now on assume that L ::::] M. The Generic Case where L ~ L 2 (2n)
(leading to the generic conclusion in our Main Theorem of a group of Lie type and
characteristic 2 of Lie rank 2) is handled in Part 2. Most cases where V is not an
FF-module for LT/02(LT) are eliminated in Part 3. The remaining cases where
Vis an FF-module are handled in Parts 4 and 5.
(^5) Recall a TI-set is a set intersecting trivially with its distinct conjugates.