1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

7.7. MINI-APPENDIX: r > 2 FOR L 3 (2).2 ON 3 EfJ 3 703

Since Zi :S V n V^9 with [V, V^9 ] -:/= 1, 3.2.10.6 says that V i. 02 (Gi). So as

IV: Ail = 2, Ai = V n 02(Gi), and hence m(V/V n 02(Gi)) = 1. Then for any

h E Gi, we have m(Vh /Vh n 02(Gi)) = 1, with Vh n 02 (Gi) :S Ti :S No(V). If

Vh centralizes V, then (V, Vh) = VVh is a 2-group, while if Vh does not centralize

V then Vh n 02(Gi) is aw-offender on V, so our argument above for Vg applies to

Vh to show (V, Vh) is again a 2-group. Therefore the Baer-Suzuki Theorem forces

V :S 02(Gi), which we saw is not the case. This completes the proof. D

7.7. mini-Appendix: r > 2 for L 3 (2).2 on 3 E9 3

Our goal in this section is to prove the following two results:

THEOREM 7.7.1. If L 0 is L 3 (2) on 3E93, then r > 2. In particular, s = m = 2.

PROPOSITION 7.7.2. Assume L 0 is L 3 (2) on 3 E9 3, and r > 2. Then

F*(Co(vi)) = 02(Ca(vi)) for each Vi E lrr+(L, V) and vi E vt.

So throughout this section, assume we are in the case where Lo is L 3 (2) on

3 E9 3. Recall LE C*(G, T), L :'SJ ME M(T), VE R2(LT) is normal in M, M :=

Mv/CM(V) ~ Aut(Ls(2)), and V =Vi E9 Vi, where Vi:= V{ fort ET-Nr(Vi)

and Vi is the dual of the natural module Vi. Recall Q := 02(LT).

The module Vis discussed in subsection H.4.1 of chapter Hof Volume I, where

we find that we can view L as the group of invertible 3 x 3 matrices over F 2 ,

with respect to some basis of Vi denoted by {1, 2, 3}, with t the inverse-transpose

automorphism.

7.7.1. Reduction to CG(V 0 ) :SM for V 0 := (1, it). Our goal in Theorem

7.7.1 is to show that r(G, V) > 2, so we need to prove that Co(U) :S M for each

U :S V with m(V /U) :S 2. It turns out this can be accomplished by controlling the

centralizer of the single subspace Vo := (1, 1 t), by showing:

PROPOSITION 7.7.3. Go:= Co(Vo) :SM.

In this short. subsection, we prove that Theorem 7.7.1 can be deduced from

Proposition 7. 7.3.

So assume Proposition 7.7.3, and suppose that for some U :S V with m(V/U) :S

2, we have Ca(U) i. M.

We first consider the case where m(V/U) = 1. Since V admits an orthogonal

form, U = v-1 for some v E V. Now replacing the orbit representatives in H.4.2 by

conjugates v = 2, 2 + 3t, 2 + 2t, we see using the form in H.4.1 that Vo :S v-1 = U,
so that Co(U) :S Co(V 0 ) :SM by Proposition 7.7.3.
Thus we have established that r > 1, so it remains to treat the case m(V/U) =
2.
First assume U is centralized by no involution of M. Then Q is Sylow in
CM(U), and no nontrivial element of odd order in M centralizes a subspace of V
of codimension 2, so that CM(U) = CM(V). Hence as r > 1, we get Co(U) :S M
from E.6.12.
This leaves the case where U is centralized by some involution 1: E M. Since
m(V/U) = 2, we mi:ist have 1: E L, and conjugating in L, we may take 1: to be given
by the matrix for the permutation (2, 3) (and hence also (2t, 3t)). So again Vo :SU,

and Proposition 7.7.3 gives Co(U) :SM.

This completes the proof of Theorem 7.7.1 modulo Proposition 7.7.3. So the
remainder of this section is devoted to the proof of Propositions 7.7.3 and 7.7.2.