1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

736 9. ELIMINATING nt(zn) ON ITS ORTHOGONAL MODULE

XE B(Gv,Tv)· Now Tacts on V1VN and there is g E NL 0 (V1VN) with vg t/:. VN

and vg E Z(Tv)· Then Vi VN s Ug, so Xg S Ca(v) = Gv, and hence Xg acts on X.

Further Tv :::; ai :::; NG ( Xg)' and x centralizes vg as vg E Viv N' so x acts on xg.

Recall from the definition of B(Gv,Tv) that X = P02(X) with P ~ EP2 or p1+^2.

Set (XXgTv)+ := XXgTv/0 2 (XXgTv)· Then Tv is irreducible on p+ /if!(P+) and

pg+ /if!(pg+), sop+ n pg+ is 1, if!(P+), or p+. As mp(XXg) s 2, the last case

holds, so x = xg. Therefore xis normal in Gv and GvY, so Lo= (LN,L~) acts

on X. Then as Aut(X/0 2 ,q,(X))= ~ SL 2 (p), either Lor Lt centralizes X/02(X),

and thus Lo= (LTv) centralizes X/0 2 (X), contradicting X = [X, LN]· This finally

establishes 9.4.1. D

LEMMA 9.4.2. (1) Ifv E Vjf, g E Lo-Na(VN), andu E Vf!l, thenCa((u,v)) S

M.

(2) Vis the unique member ofV^0 containing V1VN.

PROOF. Part (1) follows as Ca((u,v)) acts on (LN,L~) =Lo by 9.4.1. As

V1 VN = VNV).,. for suitable l EL, Ca(V1 VN) SM= Na(V) by (1). By 3.2.10.2,

M controls fusion in V, so we conclude that Na(V 1 VN) :::; M, and that (2) follows

from the proof of A.1.7.2. D

We can finally begin to implement our standard weak closure strategy.

LEMMA 9.4.3. r( G, V) > 3.

PROOF. Suppose US V with m(V/U) S 3 and Ca(U) i M. As m(V/U) S 3,

CM(U) is a 2-group by 9.1.2. Recall from 9.2.3 that r > 1, so by E.6.12, CM(U)

is a nontrivial 2-group. As m(V/U) < 4, we may take U :S Cv(t) =UN. Now for

each Vfe s UN, 1 =f=. Vfe n U as m(U N /U) s 1, so the lemma follows from 9.4.2. D.

LEMMA 9.4.4. Wo := Wo(T, V) centralizes V, so w > 0.

PROOF. Suppose A := vg :::; T with A =f=. 1. If m(CA(V)) 2: 5, then v :::;

Na(Vg) by 9.4.3, contrary to E.3.11. Hence m(A) ;?: 4, so as m 2 (M) = 4 and

'h = J(T), A= TL. Then Cv(A) = V1, so if U 1 is the £-irreducible containing V1,
then CA(L) centralizes (Vl) = U1. Now m(A/CA(L)) = 2, so as r > 3, U1 :::; Mg.
Similarly U{ s Mg, so U1U{ = V/ s Mg, and [U1Ui,A] = V1, so U 1 Uf induces
F-transvections on A with center Vj_. This is impossible since M controls fusion in
V by 3.2.10.2, while the center of an F-transvection on Vis nonsingular by 9.1.2.4,
and Vi is singular by 9.1.2.5. D

LEMMA 9.4.5. W 1 (T, V) centralizes V, sow> 1.

PROOF. If not, then arguing as in the proof of the previous lemma, there is a

hyperplane A := Vg n T of Vg with A =f=. 1, and this time m(A) 2: 3. Suppose first

Ai Lo. Then A has maximal rank (namely 3) subject to Ai L 0 , so A E A(CM(a))
for each a E A-Lo. Observe m(Vg /CA(V1)) s 2, so V1 s Mg since r > 3 by
9.4.3. Thus if A does not centralize Vi, then Z = [Vi, A] s Mn Vg = A. As
CA(Vi) is of codimension at most 2 in vg, V1 induces an F-transvection on vg

with Z contained in the center [Vg, Vi], a contradiction as in the proof of the

previous lemma. Therefore [A, V 1 ] = 1, so as A i Lo, there is t E A with t = a

and A= (t)(A n LN)· But then m(Vg /CA(UN)) s 3, so UN s Mg since r > 3.

Therefore V N V1 = [A, UN] S A S Vg, contrary to 9.4.2.2.