1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

15.3. THE ELIMINATION OF Mr/CMf(V(Mf)) = 83 wr Z 2 1145

Observe that L* is A 6 or Ln(2), 3::::; n::::; 5, since in the remaining cases in (1),

LT has no strong FF*-offenders by Theorem B.4.2, contrary to (n).

Suppose that L* ~ L3(2). As v,; ::::; 02(Y 0 *T*) and Y 0 *T* is the stabilizer of

the point Zo in Uo, v; is not a strong offender on UH by Theorem B.5.1, contrary

to (n). Thus L* is not L 3 (2).

Suppose next that L* ~ Ln(2) for n = 4 or 5. As Y 0 *T 0 is a T-invariant

minimal parabolic by (1), either LT is generated by overgroups H 1 of Y 0 T with

Hi/02(H1) ~ 83 x 83 or L3(2), or H ~ 8s with YoT 0 the middle-node minimal

parabolic of L *. In the first case, L ::::; M by our previous reductions, contrary to

H i M. In the second case, YoT = CH(Zo), so by Theorem B.5.1, U 0 is the sum

of the natural module and its dual; hence 02 (Y 0 *T*) contains no FF*-offender by

B.4.9.2iii, whereas v,; is such an offender by (h).

Thus L ~ A5. But then as VO: ::::; 02(".YQT) with "YQT* the stabilizer of the

point Zo, v; is not a strong FF*-offender on UH by B.3.2, contrary to (n). This

contradiction completes the proof of 15.3.49. D

15.3.3. The case (VMc) nonabelian. Recall from 15.3.49.4 that case (1) of

15.3.7 holds, and in particular 15.3.48 applies to all HE H(T, M).

In this subsection, we will assume that (V Mc) is nonabelian, and derive a

contradiction via an application of the methods in section 12.8; in particular we

will use Theorem G.9.3. Thus we will reduce to the following situation, to be

treated in the final subsection:

THEOREM 15.3.50. VH is abelian for each HE H(T, M).

Until the proof of Theorem 15.3.50 is complete, assume H is a counterexample.

Then (VMc) is also nonabelian, so as usual in the nonabelian case of section F.9,

we take H :=Mc. Recall Mc= Ca(Z) by 15.3.4, so VH = (VCa(Z)). Set U :=

UH= (z~a(Z)).

LEMMA 15.3.51. (1) V* =fa 1.
(2) Either
(a) U is nonabelian, U is the 4-subgroup of T distinct from S, and U is a
Sylow group of nt (V), or
(b) U is elementary abelian, U::::; 8, Z(T) ::::; U, and Zs= V n U.
(3) Y = [Y, U].
(4) [V,QH]::::; VnQH and [V,U]::::; vnu.
PROOF. If v ::::; QH then the members of vH normalize v, so that VH is
abelian by 15.3.46.4, contrary to our choice of H as a counterexample. Thus (1)
holds, so [U, V] # 1 by 15.3.48.3, and hence U =fa 1. By 15.3.48.2, (U) ::::; Z, so U
is elementary abelian, and as T ::::; H, U :::l f', so Z(T) ::::; U as Z(T) is of order

LEMMA 15.3.52. U is nonabelian.