11.2. WEAK-CLOSURE PARAMETER VALUES, AND (vNaCVil) 767H. Let B be a Hall 2'-subgroup of H n M. If A:= CB(V) =f 1, then by 4.4.13.1,
Na(A) i M, contrary to Theorem 4.4.3 using Remark 4.4.2. Thus CB(V) = 1.
Suppose that B normalizes V. Then B is faithful on V, giving Hypothesis
E.3.36~so that by E.3.38 we have n(H)::::; n'(AutM(V)) = n, contrary to assump-
tion.
Hence we may assume that B does not normalize V, so in particular V < V M.
By 11.0.3.5, L ~ SLs(q), and then by 11.0.3.4, Cv(B) = 1, so Cv(H) = 1. In
particular as Zn V =f 1, [Z, H] =f 1, so L = [L, J(T)] by 3.1.8.3. Now the argument
in the final paragraph of the proof of 11.0.3 and an appeal to B.5.1.1.ii shows VM is
the sum of two isomorphic natural modules for L ~ SLs(q). Thus CaL(VM)(L) ~GL2(q), so if m > 2n then CB(VM) =f 1, contrary to paragraph one. Thus m::::; 2n,
completing the verification of (2) and hence the proof. DRecall Mv = NM(V) = Na(V) and TL= T n L02(LT) = T n LCr(V).
LEMMA 11.2.2. Set m := 2n if L ~ G2(q) and m := n otherwise. Let U ::::; V,
and set k := m(V/U). Then
(1) m(Mv, V) = m.
{2) Assume that 0
21
(CM(U))::::; CM(V) and k < 2m. Then Ca(U)::::; M, andso 0
21
(Ca(U)) ::::; CM(V).
{3) Either r(G, V) > m; or L ~ SL3(q), r(G, V) = m, and Ca(Vz) i M. Inparticular, s(G, V) = m.
(4) Wj(T, V)::::; TL for j < m -1, so Vi::::; Cv(Wj(T, V)).
(5) If L ~ G 2 (q) then Wo(T, V) ::::; Cr(V), so Na(Wo(T, V)) ::::; M; that is,w(G, V) > 0.
(6) If L ~ G2(q) then Ca(C1(R1, V))::::; M.
PROOF. Part (1) is a standard fact about the natural module and its nonsplit
central extensions in I.2.3.1; cf. B.4.6 when L ~ G2(q).
Next we claim r(G, V) ;:::: m. By (1), m(Mv, V) 2 m; so if m > 2, the claim
follows from Theorem E.6.3. Assume m::::; 2; then m = 2 = n, and Lis SL 3 (4) or
Sp 4 (4). If L is Sp 4 (4), assume further that Cv(L) = 1. Then L is transitive on
non-zero vectors in the dual of V, and hence transitive on Fr hyperplanes U of V,so in particular each hyperplane is invariant under a Sylow 2-subgroup of M. Hence
as m(Mv, V) = m;:::: n > 1 by (1), r(G, V) > 1 by E.6.13. Thus we may assume
L/0 2 (L) ~ Sp4(4), Cv(L) =f 1, and U is a hyperplane of V with Ca(U) i M. By
Theorem 4.2.13, M = !M(L), so Cu(L) = 1; hence U is an F 2 -space complement
to Cv(L), and so m(Cv(L)) = 1. Now Vis a quotient of the full covering V of the
natural module V for L, which has the structure of a 5-dimensional orthogonal space
over F 4. From this structure, L has two orbits on the F4-complements to Cy-(L)
in V, with representatives Ue, E = ±1, such that AutL(Ue) ~ 01(4). Moreover
each F 2 -hyperplane of V supplementing Cy-(L) contains such an F4-hyperplane,so the images Ue of Ue, for E = ±1, are representatives for the orbits of L on
F 2 -complements.to Cv(L). In particular NLr(U) is maximal in LT but not of
index 2, and there is a subgroup Y of order 3 in NL(U) faithful on U. Next
Zn Vi Cv(L), so that Zn U =f 1, and hence Na(U) E 7-le by E.6.6.4. By E.6.7.1,
Ca(U) contains ax-block invariant under Y = 02 (Y). Then as Y is faithful on U,
while m 3 (YCa(U))::::; 2 as Mis an SQTK-group, ms(Ca(U))::::; 1. Hence we have