754 10. THE CASE LE .Cj(G, T) NOT NORMAL IN M.
Assume (3) fails. If V centralizes Vz, then as Caz (Vz) :::; M by (1), V :::;
02 (Ccz(Vz)) :::; 02 (Gz), contrary to assumption. Hence as XT is irreducible on
V/ E, E = Cv(Vz). If X '.':] Gz, then as X centralizes E, it centralizes Vz; then
V = [V, X]E centralizes Vz, a contradiction. Thus X < Kz, and hence Kz i. M so
Vz E R2(Gz) by (2).
As E = Cv(Vz), Vt is a 4-group. By (1), Vz :::; NM(Vi), so [Vz, Vi] _:::; Vz n
V 1 = (v 1 ). That is Vt is a 4-group inducing transvections on Vz with center v1.
Further K;T is described in case (2) or (3) of 10.2.4. Appealing to G.3.1, the
only group K;T listed there containing a 4-group of F2-transvections with a fixed
center in some representation is L3(2) wr Z2 with [Vz, Kz] = Vz,1 EB Vz,2, where
Vz,i := [Vz, Kz,i] is a natural module. However in that case, Vi* :::; K;,i with
vi= [Vz, Vi*]:::; Vz,i, so z = V1V2 E [Vz,Kz], which is impossible as z E Z(Gz) but
C[Vz,KzJ(K;) = 1. This contradiction completes the proof. D
We can now prove our major weak closure result, which establishes an effective
lower bound on the parameter w ( G, V).PROPOSITION 10.2.12. One of the following holds:
(1) w(G, V) > 2.
(2) w(G, V) = 2, and case (3) of 10..1.1 holds.
(3) w(G, V) = 2, and case (1) of 10.1.1 holds with n = 2.PROOF. In case (3) of 10.1.1, and in case (1) when n = 2, set j := 1. Otherwise
set j := 2. We must prove w(G, V) > j, so we may assume A := Vg n M with
k := m(Vg /A) :::; j and [V, Vg] -/= 1, and it remains to derive a contradiction.
Let m := m(V1) and a := a(AutM(Vi), V1). Observe m > j + 1. Recall
a :::; m 2 (AutM(Vi)) and in case (2) of 10.1.1, a= 1. Thus k < m - a unless case
(3) of 10.1.1 holds and k = 1.
For i = 1, 2, set Ai :=Vig nA and Bi:= NAi (Vi). Suppose AiA2 centralizes V1.
Then by 10.2.9.1, Vi :::; NM9 (Vig), so Cv 1 (Vig) -/= 1 since m(Vi) < m2(AutM(V1))in each case. Then A = Vg by another application of 10.2.9.1. But then Vg =
A 1 A2:::; CM(V1) = CM(V), contrary to our choice of Vg. Thus we may assume Ai
does not centralize V1 for some choice of i := 1 or 2.
Next m(Vig /Ai) :::; k with m(Ai/Bi) :::; 1, so m(Vig /Bi) :::; k + 1 < m =
m(Vig /Cv.B(L6)) by paragraph two. Thus Bi i. Cvg(L6), so there exists b E Bi -
Cv9 (Lg). 'For each such band each;= 1, 2, we may apply 10.2.9.1 to getCvr (b) _:::; Nvr (Vig) =: Un
so Vo := [Ai, Cv 1 (b)] :::; Vig n V and [Bi, CVi (b)] :::; Vig n V1 = 1by10.2.10.1. Thus if
Vo -/= 1 then Ai > Bi and V:::; Cc(Vo) :::; Mg by 10.2.9.1. Thus [A, VJ :::; Vg n V:::;
Cv(b), and as A> Bi, for any a E Ai - Bi, V = [a, V]V2, sob centralizes V/V 2.Thus b E CT(V/V2) = CT(V1), so V1 = Cv 1 (b) and V = V 0 \12. Then by 10.2.9.1,
L = (CL(vo): Vo E vet):::; Mg,
so Lo= (LAi) :::; Mg and hence Lo= Lg by 10,1.3, contradicting g ¢:. M. Therefore
Vo= 1, so
As Cv 1 (b) i. Cv 1 (L) from the structure of the modules in 10.1.1, Ai acts on V1 by
(*), so Ai =Bi. Then as Ai does not centralize Vi, (*) says