2.2. BENDER GROUPS 519PROPOSITION 2.2.2. Assume for each D in the Alperin-Goldschmidt conjuga-tion family that Na(D) ::; M for T in G. Then
(1) Each normal 2-subgroup of M is strongly closed in T with respect to G.
(2) G is a Bender group.PROOF. We first prove (1). Let Ube a normal 2-subgroup of M, u E U and
g E G with ug ET. We must show ug EU, so assume otherwise. By the Alperin-
Goldschmidt Fusion Theorem (the elementary result 16.1 in [GLS96], proved as
X.4.8 and X.4.12 in [HB85]), there exist u =: u1, ... , Un := ug in T, Di E 'D, and
Xi E Na(Di), 1::; i < n, such that ug = ux^1 ···xn-^1 , (ui,UH1)::; Di, and u~i = Ui+l·As u = u1 E U but Un = ug ¢:. U, there exists a least i such that ui+ 1 ¢:. U.
Thus ui E U, and by hypothesis Xi E Na(Di) ::; M; therefore as U :s! M, also
ui+l = u~' EU, contrary to the choice of i. Thus (1) holds.
We could now appeal to Goldschmidt's Fusion Theorem [Gol74] to establish
(2). However the version of this theorem in our list of Background References(cf. Theorem SA in [GLS99]) assumes that G is of even type, whereas in the
Main Theorem we assume G is of even characteristic. Fortunately the even type
hypothesis is unnecessary, and we now extract an easier version of the proof from
section 24 of [GLS99] under our own hypotheses:Let U be a minimal normal 2-subgroup of M. Then U is elementary abelian,
Mis irreducible on U, M = Na(U), and U is strongly closed in G by (1). Thus for
u E u#' u^0 n M ~ u and M controls fusion in u by Burnside's Fusion Lemma, sou^0 n M = uM. Set Gu:= Ca(u).
As U :s! T, we may choose z E Z(T) n U#. Hence Gz ::; Mas M = !M(T),
so as z^0 n M = zM, M is the unique point fixed by z in the representation of G by
right multiplication on the coset space G/M (cf. 46.1 in [Asc86a]). We use this
fact to show:(*) For each 2-subgroup S of G containing z, Na(S) ::; M.
For C 8 (z) fixes the unique fixed point M of z on G / M, and hence M is the unique
fixed point of Cs(z) on G/M. Then as each subgroup of Sis subnormal in S, we
conclude by induction on [Sf that Mis the unique fixed point of S of G/M. Hence
Na(S)::; M.
First assume Gu ::; M for every u E U#. Then as U is not normal in G, RemarkI.8.4 and Theorem I.8.3 tell us that G is a Bender group.
Thus we may assume that :J := { u E U# : Gu 1:. M} is nonempty, and
it remains to derive a contradiction. In particular U > (z), so the elementary
abelian group U is noncyclic. Let u E :!, set H :=Gu, MH :=Mn H, and let
U::; SE Syl 2 (H). By(*), S::; M, so conjugating in M we may assume that S::; T.
By 1.1.6 applied to the 2-local Gu = H, the hypotheses of 1.1.5 are satisfied, so
MH E 7-ie by 1.1.5.l.
Suppose H E 7-ie. Then as S ::; T and S E Sy[z(H), z E Z(S) ::; 02 (H), so
H::; Na(02(H)) ::; M by (*), contradicting Hf:. M. Thus Hf:. 7-ie.
Let W be any hyperplane of U. Then [zM n U/ > 1 as U is noncyclic, so
zM n W =f. 0 by A.l.43. Now as Gz ::; M, Ca(W) ::; Ca(zM n W) ::; M. Hence
using Generation by Centralizers of Hyperplanes A.l.17, O(H) = (Co(H)(W) :
m(U/W) = 1)::; M, so O(H) = 1 since MH E 7-ie.
Thus as Hf:. 7-ie, there is a component L of H, and by 1.1.5, L = [L,z] 1:. M
and Lis described in 1.1.5.3. Set Lo := (LH). As U is strongly closed in S with