518 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1
2.1. Statement of main result
Our main theorem in this chapter is:
THEOREM 2.1.1. Assume G is a simple QTKE-group, TE Syb(G), and M =!M(T). Then G is a Bender group, L 2 (p) for p > 7 a Fermat or Mersenne prime,
L3(3), or Mu.
Of course the groups appearing in the conclusion of Theorem 2.1.l also appear
in the conclusion of our Main Theorem. Thus after Theorem 2.1.1 is proved, we
will be able to assume that IM(T)I;::: 2 in the remainder of our work.
Throughout chapter 2, we assume that G, M, T satisfy the hypotheses of
Theorem 2.1.1. Thus M = !M(T), and hence by Sylow's Theorem, also M =
!M(T') for each Sylow 2-subgroup T' of M, so we are free to let T vary over
Syl2(M).2.2. Bender groups
As we mentioned, the generic examples in Theorem 2.1.1 are Bender groups.
These groups were originally characterized by Bender as the simple groups G with
the property that the Sylow 2-normalizer Mis strongly embedded in G; that is (cf.
I.8.1), Na(D) ~ M for all nontrivial 2-subgroups D of M.
If we assume that G is not a Bender group, then there is 1 < D ~ T with
Na(D) i M, so that Na(D) E H(D, M) in our notation. If we pick D so t:p.atU := NT(D) is of maximal order subject to this constraint, then since M = !M(T)
by hypothesis, U is a proper subgroup of T Sylow in Na(D) with Na(U) ~ M.
Our proof will focus on pairs (U, Hu) such that U ~ T, U E Syb(Hu ), and
Hu E He(U, M). While the pair (U, Na(D)) satisfies the first two conditions, and
Na(D) E H(U, M), it may not be the case that Na(D) E He. Thus to ensure that
such pairs exist, we use an approach due to GLS (cf. p. 97 in [GLS94]) to produce
a nontrivial strongly closed abelian 2-subgroup in the absence of such pairs. Thenwe argue as in the GLS proof^1 of Goldschmidt's Fusion Theorem, to show that G
is a Bender group. Our extra hypotheses makes the proof here much easier. We
identify G using a special case of Shult's Fusion Theorem, which appears in Volume
I as Theorem J[.8.3, and is deduced in Volume I from Theorem ZD in [GLS99]. ·We now begin to implement the GLS approach. Instead of considering arbitrary
subgroups D of T, we focus on the members of the Alperin-Goldschmidt conjugation
family: Using the language of Theorem 16.1 in [GLS96] (a form of the Alperin-
Goldschmidt Fusion Theorem):
DEFINITION 2.2.1. Given a finite group G and TE Syl 2 (G), define V to be the
set of all nontrivial subgroups D of T such that
(a) NT(D) E Syb(Na(D)),
(b) Ca(D) ~ 021, 2 (Na(D)), and
(c) 021, 2 (Na(D)) = O(Na(D)) x D.
The set V is called the Alperin-Goldschmidt conjugation family for Tin G.
Next recall that a subgroup X of Tis strongly closed in T with respect to G if
for each g E G, Xg n T ~ X.(^1) See the proof of Theorem SA in section 24 of [GLS99]-but recall that we will not make
use of their hypothesis of even type.