CHAPTER 2
Classifying the groups with IM (T) I = 1
Recall from the outline in the Introduction to Volume II that the bulk of the
proof of the Main Theorem proceeds under the Thompson amalgam strategy, which
is based on the interaction of a pair of distinct maximal 2-local subgroups containinga Sylow 2-subgroup T of G. Clearly before we can implement that strategy, we must
treat the case where Tis contained in a unique maximal 2-local subgroup.
In Theorem 2.1.1 of this chapter, we determine the simple QTKE-groups Gin
which a Sylow 2-subgroup Tis contained in a unique maximal 2-local subgroup.
This condition is similar to the hypothesis defining an abstract minimal par-
abolic B.6.1, where T lies in a unique maximal subgroup of G, so we can expect
many of the examples arising in E.2.2 to appear as conclusions in Theorem 2.1.1.
The generic examples of simple QTKE-groups with IM (T) I = 1 are the Bender
groups. Recall a Bender group is a simple group of Lie type and characteristic 2of Lie rank 1; namely L 2 (2n), Sz(2n), or U 3 (2n). The Bender groups also appear
in case (a) of E.2.2.2. In addition, some groups from cases (c) and (d) of E.2.2.2
also satisfy the hypotheses of Theorem 2.1.1, as does Mn which is not a minimal
parabolic.
However, shadows of various groups which are not simple also intrude, and
eliminating them is fairly difficult. We mention in particular the shadows of certain
groups of Lie type and Lie rank 2 of characteristic 2, extended by an outer automor-
phism nontrivial on the Dynkin diagram: namely as in cases (la) and (2b) ofE.2.2,extensions of the groups L 2 (2n) x L2(2n), Sz(2n) x Sz(2n), L3(2n), and Sp4(2n).
. These groups are not simple, but they are QTKE-groups with the property that
the normalizer of a Borel subgroup is the unique maximal 2-local containing a Sy-
low 2-subgroup. We will eliminate the first two families of shadows in 2.2.5 by
first using the Alperin-Goldschmidt Fusion Theorem to produce a strongly closedabelian subgroup, and then arguing that G is a Bender group to derive a contra-
diction. However it is difficult to see that the shadows of the latter two families
are not simple, until we have reconstructed in Theorem 2.4. 7 most of their local
structure, and are then able to transfer off the graph automorphisms and so obtain
a contradiction.
Also certain groups of Lie type and odd characteristic are troublesome: Thegroups L 2 (p) x L 2 (p), p a Fermat or Mersenne prime, extended by a 2-group
interchanging the components (a subcase of case (b) of E.2.2.l); and the groupL 4 (3) ~ Pnt(3) extended by a group of automorphisms not contained in PO;t(2).
These groups are also minimal parabolics but. not strongly quasi thin. Shadows
related to the last group appear in many places in the proof.
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