m(Q/Q 1 ) ·:::::; m(H^1 (L, V)), so H^1 (L, V) -=/= 0. Therefore by I.1.6, L is an Sp4(4)-
block and m(Q/Q 1 ):::::; 2, and by I.2.3.1, Q/CT(L) is a submodule of the dual of the
natural 5-dimensional module over F 4 for Ds(4) ~ Sp4(4). Here we compute (e.g.
by restricting to the subgroup Sp4(2) ~ 85) that Cq(R1) :::::; Qi and Cv(R1) = V1.
Therefore Z1 := D1(Z(R1)) = V1Cz 1 (L).
If CT(L) = 1 then Z 1 = Vi by the previous paragraph, and hence (2) holds.
Thus we may assume that CT(L)-=/= 1. Now [02(LT), X]:::::; [02(LT), L] = V, so
Z1 := D1(Z(R1)) = Zx x Zc,
where Zx := [V1,X], Zc := Cz 1 (X) = Cz 1 (L), and Zc-=/= 1 as CT(L)-=/= 1. For
D :::::; G, let B(D) be the subgroup g~nerated by all elements of D whose order lies
in .6., where ,6. is the set of divisors of 2n - 1 if L is SL 3 (2n) with n odd, and
,6. := {3} otherwise. Thus L = B(M) by 11.0.4. By Theorem 4.2.13, M = !M(L),
so Zc is a TI-set under the action of Y := Na(R 1 ), with YM ::;= YnM = Ny(Zc):
for if y E Y with 1-=!= Zc n Z6, then (L,LY):::::; Ca(Zc n Z'b), so as M = !M(L),
LY:::::; B(Ca(Zc n Z'b)):::::; B(M) = L, and hence y E Ny(L) = YM. Notice YM < Y
as (1) fails. Set Y := Y/Cy(Z 1 ). Then X is regular on zJ::., and normal in
Y.M since L 1 R 1 centralizes V 1. Thus we have the hypotheses for a Goldschmidt-
O'Nan pair in the sense of Definition 14.1 in [GLS96]; so we may apply O'Nan's
Lemma 14.2 in [GLS96, 14.2], with Y, X, Z 1 in the roles of "X, Y, V". Observe
conclusion (iv) of that result must hold-since in (i), Y normalizes Zc giving YM =
Y; while in (ii) and (iii), T does not normalize Za. In conclusion (iv) of 14.2 of
[GLS96, 14.2], q = 4, Z 1 ~ E 8 , and Y* is a Frobenius group of order 21. NextCa(Z 1 ) :::::; Ca(Zc) :::::; M = !M(L), so L1 E C(Na(Z1)) and hence Y acts on L1.
If L is an SL 3 (4)-block, the noncentral 2-chief factors for L 1 are VZ(L1)/Z(L1)
and 02 (L1)/VZ(L 1 ), and both are natural modules. Therefore the induced action
of Na(L1) on Irr+(L 1 ,0 2 (L 1 )/Z(L 1 )) is contained in rL2(4), so 071 (Y) acts on
VZ(L 1 ) and then on [VZ(L 1 ),L1] = V. But then Y = 07 ' (Y)YM:::::; Na(V) s M,
contradicting YM < Y. Similarly if Lis an Sp4(4)-block, then Y acts on [V,L 1 ] =
Vi, so Y :::::; JV[ by 11.1.4, for the same contradiction. This completes the proof. D
11.2. Weak-closure parameter values, and (vNa(V^1 ))
Since Vis an FF-module, we do not have the ideal situation for weak closure
described in subsection E.3.3; however, we will be able to establish at least some
restrictions on the weak closure parameters r(G, V), w(G, V), and n(H) discussed
in Definitions E.3.3, E.3.23, and E.1.6. Recall that the paramter n'(AutM(V)) is
defined in Definition E.3.37, and notice that n'(AutM(V)) = n > 1: for example
this follows from A.3.15.
LEMMA 11.2.1. For HE 'H*(T,M), either
(1) n(H) s n, or
(2) L ~ SL3(q), VM is the sum of two isomorphic natural modules for L/02(L),
Cv(H) = 1, L = [L, J(T)], and n(H) s 2n.^2
PROOF. Assume (1) fails, so that n(H) > n > 1. Then by E.2.2, 02 (H/0 2 (H))
is of Lie type over F 2 m, form := n(H) > n, and H n Mis a Borel subgroup of
(^2) Notice this essentially eliminates the shadow of D8(2n), in which n(H) = 2n but V :SJ M.
Our use of the quasithin hypothesis is via reference to the pushing up result Theorem 4.4.3.