2.3. PRELIMINARY ANALYSIS OF THE SET ro 521PROOF. Let 1) be the Alperin-Goldschmidt conjugation family for Tin G. By
2.2.3, D ~ S2(G). Therefore if o = 0, then Na(D) ::::; M for each DE 1>. Hence by
Proposition 2.2.2.2, G is a Bender group. DREMARK 2.2.6. The idea of using the Alperin-Goldschmidt Fusion Theorem
and Goldschmidt's Fusion Theorem in this way is due to GLS. This approach allows
us to avoid considering the case where the centralizer of some involution i has a
component which is a Bender group: For if i is such an involution then U (Ca( i)) = 0
(in the language of Notation 2.3.4 established later), whereas Theorem 2.2.5 allows
us to assume o =/= 0, which supplies us with 2-locals H such that U(H) =/= 0.
It is these 2-locals which we will exploit during the. remainder of this chapter.In particular, as mentioned in the introduction to the chapter, this allows us to
avoid difficulties with the shadows of Bender groups extended by involutory outer
automorphisms, and also with the shadows of the wreathed products L2 (2n) wr Z2and Sz(2n) wr Z 2.
2.3. Preliminary analysis of the set I' 0
Since the Bender groups appear in the conclusion of Theorem 2.1.1, by Theorem
2.2.5, we may assume for the remainder of this chapter that
0 =/= 0, so that also o* =/= 0.
Recall from the second paragraph of the previous section that there exist pairs
(U, Hu) such that U E Syl 2 (Hu ), Na(U) ::::; M, and Hu E 'H(U, M). Using the fact
that o is nonempty, we will produce such pairs. with Hu in 'He(u, M). Moreover we
will see that we can choose U to have a number of useful properties which we list
in the next definition:
NOTATION 2.3.1. Let f3 = f3M consist of those U E S2(G) such that
(/3 0 ) U :S: M, so in fact U E S2(M);
(/31) For all U :S: VE S2(M), Na(V) :S: M; and
(/32) Co 2 (M) (U) :S: U.
Notice that (/3 0 )-(/3 2 ) are inherited by any overgroup of U in S2(M), so all such
overgroups are also in f3. Some other elementary consequences of this definition
include:
LEMMA 2.3.2. Assume U E /3, and U :S: H :S:: G. Then
(1) If U ::::; V E S 2 (G), then V E /3. In particular all 2-overgroups of U in Glie in M.
(2) IHl2 = IH n Ml2·
(3) If HE 'He, then 02(H) E S2(G). In particular /3 ~ S2(G).
PROOF. To prove (1), assume U::::; VE S 2 (G). Recall that each 2-overgroup
V of U in M is in /3, so it only remains to show that V ::::; M. If U :::;I V, then
v ::::; Na(U) ::::; M by (/31). So as u :::;I :::;I v, v ::::; M by induction on IV : u1,
completing the proof of (1).
Next let U::::; SE Syl 2 (H). Then SE /3 by (1), so S :S: M by (/30), giving (2).
Finally set Q := 02 (H), so that Q ::::; S since SE Syb(H). As S :S: M, we. may
assume that S :S: T. Then Ch(M) :S: T as TE Syl2(M), so Z(T) :S: Co 2 (M) (S) :S:
Z(S) by (/3 2 ). Under the hypothesis of (3), Q = F*(H), so Z(T) ::::; Z(S) :S::