1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

15.2. FINISHING THE REDUCTION TO Mf/CMf(V(M£)) ~ o:t(2) 1119

PROOF. We claim that 03 ,(Mc) :::;: Mn Mc: for otherwise we may choose a

T-invariant 3'-subgroup K of Mc minimal with respect to J := KT i M; then

J E H (T, M), whereas members of H (T, M) are not 3' -groups by 15.2.8. So the

claim holds, and hence as 031 (Mc) =Kc by 15.2.12.4, (1) holds.

If (2) fails, then [Kc, VJ -=/-1 by (1), so K < Kc as V :::;: 02(H) by 15.1.11.1.

Thus case (ii) or (iii) of 15.2.12.2 holds. and then by Kc = [Kc, VJ by (1). Let

QC := 02(Mc), v;, := v n Qc, K;T* := KcT/02(KcT), and Mc := Mc/Z. Then

CMJQc) :::;: Qc by A.1.8. Thus as V* -=/-1, V does not centralize Qc, so as [Qc, V] :::;:

Ve, m(Vc) ~ 1. On the other hand since m(Z) ~ 1 :::;: m(V*) and V has rank

4, m(Vc) :::;: 2 with equality holding only if V* and Z are of rank 1. Next in the

groups in (ii) and (iii) of 15.2.12.2, no normal subgroup of T* induces a group of

F2-transvections with fixed center on a chief section of Qc by B.4.2, keeping in mind
in (ii) that T is nontrivial on the Dynkin diagram of K;. Therefore we conclude
that [V, Q c] = ii;, is of rank 2, so that V and Z are indeed of order 2. It follows that
V = Z(T), KcT has just one noncentral 2-chief factor W, and (e.g., by D.3.10,
B.4.2, and B.4.5) either
(i) K;T ~ 88 or Aut(L 5 (2)), and either W is the 6-dimensional orthogonal
module for 88 , or Wis the sum of the natural module for K; ~ Ln(2) (n = 4 or
5) and its dual; or
(ii) K;T ~ L3(2) wr Z2 and W = W1 E9 W2, where Wi := [W, Kc,i] is the
natural module for the direct factor K;,i ~ L3(2) of K;.
Next as Z is of order 2 and Zs is of order 4, 1 -=f. Zs. As Zs :::;: Z(T), we

conclude Zs:::;: ~V =ii;, using B.2.14. Thus the projection Wz of Zs on Wis

nontrivial and centralized by H by 15.2.11.2. As H ~ 83 wr Z2, it follows that

in (i), K;T ~ 88 and Wis the orthogonal module; and in (ii), H is the parabolic
of K;T over T stabilizing a point of W. In either case, there is a parabolic
P of K;T not contained in H, minimal subject to being T -invariant; further
P /0 2 (P) ~ 83 in (i), and P ~ 83 wr Z 2 in (ii). By minimality of P, if the
preimage Pis not contained in M, then PE H*(T, M). We conclude from 15.2.8
that P:::;: Min (i), while in (ii) we get P:::;: M from 15.2.11.2 since [Wz,P] -=f. 1
by construction. Then by 15.2.14.5, 02 (P) :::;: Cp(V) :::;: Cp(Zs), again contrary to
[Wz, P] -=f. 1. This contradiction completes the proof of (2).
Now by (2), V"' :::;: Qc :::;: T for each x E Mc, so [V, V"'] = 1by15.2.26.2. Finally

assume that 1 -=f. Zs n V9 for some g E G. As V:::;: Z(J(T)) and Na(J(T)) :::;: M

by 15.1.9.1, we may apply Burnside's Fusion Lemma A.1.35 to conclude that M

controls fusion in V. Therefore we may take g E Na(Zs n V^9 ) by A.1.7.1, and

hence g E Mc by Theorem 15.2.15. Then [V, V9] = 1 by the initial remark of the

paragraph, so (3) holds. D

LEMMA 15.2.28. [Us,A(Qs)]:::;: Zs.

PROOF. Observe that A( Q H) :::;: A(T) :::;: 8 by 15.2.25.2, and 8 centralizes

V/Zs in each case of 15.2.25.1. Thus as A(Qs) :s;J Hand Us= (VH), the lemma
holds. D

We are now in a position to complete the.proof of Theorem 15.1.3.

Observe that the pair M, H satisfies Hypotheses F.7.1 and F.7.6 in the roles

of "G 1 , G 2 ". Form the coset graph r on the pair M, H, and adopt the notation
of section F.7. In particular /'o and /'l are the points of r stabilized by Mand H,