1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

4.4. CONTROLLING SUITABLE ODD LOCALS 623

In cases (d)-(f), VB is a natural module for LB/0 2 (LB), so that subcase (i) of case

(2) of Theorem 4.4.3 holds, contrary to our assumption that B affords a counterex-

ample to Theorem 4.4.3. Hence it only remains to dispose of cases (a)-(c).

Assume first that case (b) holds. Then from the structure of K ~ M 24 or He,

VB i 02(CK(z)) for each z E CvB (T n L)#. Hence VB i 02(Cc(z)), so condition

in (b) of subcase (i) of case (2) in Theorem 4.4.3 holds, again contrary to our choice

of a counterexample. Similarly if case ( c) holds then from the structure of Ru

(cf. the case corresponding to Ru in the proof of Theorem 4.2.13, using facts from

chapter J) of Volume I, CK(Vs) i MK. Thus condition (c) of subcase (i) of case

(2) in Theorem 4.4.3.2 holds, for the same contradiction.

Therefore we may assume case (a) holds. Set ZB := CvB(TB) and Gz :=

Cc(ZB). Observe that ZB is of order 2 and Kz := CK(ZB)^00 ~ M 22 /2^1 +1^2.

Arguing as in the last paragraph of the proof of Theorem 4.2.13, T induces inner

automorphisms on L/02(L), and hence LT = LR+; therefore as VB :S: Z(R+),

ZB :S: Z(T), so T :S: Gz. By 1.2.1.1, Kz is contained in the product of the

members of C(Gz) on which it has nontrivial projection. Since m 3 (Kz) = 2 and

Gz is an SQTK-group, there is just one such member, so that Kz :S: Lz E C(Gz),

and from 1.2.1.4, Lz/0 2 (Lz) is a quasisimple group described in Theorem C. Set

(LzB)* := LzB/CLzB(Lz/02(Lz)).

Then Kz E C(CL'Z(B*)) with K'f/02(K'f) ~ M22 or M22. Inspecting the p-locals

(for odd primes p) of the groups in Theorem C, we conclude that either Kz = LZ.

or LZ. ~ J4 and B* = Z(K'Z) is of order 3. In the latter case, Kz. :S: Iz :S: Lz

with Iz E £( G, T) and Iz ~ M 22 /21+^12. Thus replacing Lz by Iz in this case,

and replacing the condition that Lz E C(Gz) by Lz E C(G,T), we may assume

Lz = Kz02(Lz).

Thus in either case, Lz E C(G, T) with Lz = Kz0 2 (Lz) and [Lz, BJ :S:

02 (Lz). Let X := (BT); then X = 02 (X) = 02 (XT). As [L, B] :S: 02(L),

[L,X] :S: 02 (L) :S: T :S: Nc(X), so that X = 02 (X0 2 (L)) :::1 LTX, and hence

Nc(X) :S: M = !M(LT). Similarly as [Lz, B] :S: 02(Lz), Lz :S: Nc(X), and hence

Kz :S: LzT :S: Nc(X). Now K = (LB, Kz) :S: Nc(X) :S: M, contradicting 4.4.10.1.

This final contradiction completes the proof of Theorem 4.4.3.

We interject a lemma which is often used in applying Theorem 4.4.3. Recall

the notation n(H) in Definition E.1.6.

LEMMA 4.4.13. Assume that G is a simple QTKE-group, HE 1-l with n(H) >

1, SE Syl 2 (H), and Sis contained in a unique maximal subgroup MH of H. Then

MH n 02 (H) is 2-closed, and if we let B denote a Hall 2'-subgroup of MH, then:

(1) If A is an elementary abelian p-subgroup of B with AS = SA, then H =

(MH, NH(A)). In particular NH(A) i MH.

(2) Assume that ME M(S), MH = MnH, and M+ = 02 (M+) :::1 M. Then

CB(M+/02(M+))S = SCB(M+/02(M+)).

PROOF. As n(H) > 0, Sis not normal in H, so as MH is the unique maximal
subgroup of H over S, His a minimal parabolic in the sense of Definition B.6.1.
As n := n(H) > 1, E.2.2 then says that Ko := 02 (H) = (K^8 ) for some K E C(H)
with K/0 2 (K) a Bender group over F 2 n, (S)L 3 (2n), or Sp4(2n), and in the latter
two cases Sis nontrivial on the Dynkin diagram of K/02(K). Set H* := H/0 2 (H)
and M 0 := MH n K 0. By E.2.2, Mo is the Borel subgroup of Ko over Sn Ko. In