0.3. AN OUTLINE OF THE PROOF OF THE MAIN THEOREM 487There have since been new treatments of portions of the N-group problem due
to Stellmacher [Ste97] and to Gomi and his collaborators [GT85], using an ex-
tension of the Tutte-Sims theory which has come to be known as the amalgam
method. The Thin Group Paper [Asc78b] used some early versions of such exten-
sions due to Glauberman, which eventually were incorporated in the proof of the
Glauberman-Niles/Campbell Theorem [GN83]. Goldschmidt initiated the "mod-
ern" amalgam method in [Gol80], and this was extended and the amalgam method
modified in [DGS85] by Goldschmidt, Delgado, and Stellmacher, and in [Ste92]
by Stellmacher. Those techniques and more recent developments are used in places
in this work; our approach is a bit different from the standard approach, and is
described briefly in section 0.10 of the Introduction to Volume I.0.3. An Outline of the Proof of the Main Theorem
In this section we introduce some fundamental concepts and notation, and givea rough outline of the proof of the Main Theorem. Throughout the section, assume
G is a simple QTKE-group and T E Syb ( G). Recall that M is the set of maximal
2-local subgroups of G, and M(T) is the collection of maximal 2-locals containing
T.
0.3.1. Setting up the Thompson amalgam strategy. An overall-strategy
for studying groups of even characteristic originated in Thompson's N-group paper
[Tho68]; generically it involves exploiting the interaction of distinct maximal 2-locals M, N E M(T). (We sometimes· refer to this as the "Thompson amalgam
strategy").
Of course prior to this generic case, we must first deal with the "disconnected"
case where T lies in a unique maximal 2-local. To indicate that [M(T)[ = 1, we will
usually write :3!M(T), to emphasize the existence of the unique maximal 2-local
overgroup of T. Recall that in the generic conclusion of the Main Theorem, where
G is of Lie type of Lie rank at least 2, there are distinct maximal parabolics above
T. So for us, the disconnected case will have as its generic conclusion the groups
of Lie type of characteristic 2 and Lie rank 1. We handle this in Theorem 2.1.1,
which says:
Theorem 2.1.1 If G is a simple QTKE-group such that :3!M(T), then G is
a rank 1 group of Lie type and characteristic 2, L 2 (p) with p > 7 a Mersenne or
Fermat prime, L3(3), or Mn.
A finite group G is disconnected at the prime 2 if the commuting graph on
vertices given by the set of nonidentity 2-elements of G (whose edges are pairs of
vertices which commute as subgroups) is disconnected. The groups of Lie type and
characteristic 2 of Lie rank 1 are the simple groups of 2-rank at least 2 which are
disconnected at the prime 2. The classification of these groups is due to Bender
[Ben71] and Suzuki [Suz64); indeed the groups (namely L 2 (2n), Sz(2n), U 3 (2n))
are often referred to as Bender groups. However when working with groups of even
characteristic, a weaker notion of disconnected group is also important: namely a
group G of even characteristic should be regarded as disconnected if :3!M(T) for
TE Syh(G).
In view of Theorem 2.1.1, henceforth we will assume that [M(T) [ 2: 2. Thomp-
son's strategy now fixes a particular maximal 2-local ME M(T). Then instead of
working with another maximal 2-local, it will be more advantageous (for reasons