5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 641where Wi = V, W2 =Un U^1 , for some l EL - Gi, and W/W 2 is the sum of r
natural modules for L/02(L) and some 0 :::; r, with (Un W)/W2 = Cw;w 2 (U).In particular W = [W, DL]W2. But DL :::; CK and CK is a solvable 3'-group, so
by A.l.26.2, [W, DL] :::; 02(CK) ::; 02(M+) ::; 02(Hi) =Qi using A.1.6. Thus as
W2::; U::; Qi,
W :::; Qi ::; Cc(U).
Therefore as Zi ::; W2 ::; U n W and (U n W) /W 2 = Cw;w 2 (U), it follows that
W :::; U. But in G.2.3.6, (Un W)/W 2 is a proper direct summand of W/W 2 ifr > 0, so we conclude W = W2 and thus [0 2 (1), J] ::; W 2. Then as L :::; I and
[W2, L] = V, we conclude V = [02(L), L], so that L is an L 2 (2n)-block, contrary
to an earlier observation.
This contradiction shows that U is elementary abelian. Applying this result to
Gi in the role of "Hi", we conclude that (V^01 ) is abelian. But L is transitive on
V# and Zi is a TI-set in G, so (cf. A.l.7.1) Gi is transitive on {V9 : Zin V9 -=fa 1},
and hence as (V^01 ) is abelian, [V, V9] = 1 whenever Zi n V9 -=fa l. This verifies part
(a) of Hypothesis F.8.1 with Zi, HDL in the roles of "Vi, H".
During the remainder of the proof take Hi := HDL· Then part (b) of Hy-
pothesis F.8.1 is part of Hypothesis G.2.1 verified earlier. Next using 3.1.4.1,
CH 1 (if) :::; NH 1 (V) = Hi n M = TBDL. As V is the natural module for L,
CNaL(vJ(L)(V) ~ Z2n-i, so as DL is aHall subgroup ofTBDL and DL is faithful on
V, we conclude CH 1 (if)= CTB(V). Therefore ker 0 H 1 ("V")(Hi) :::; kerTB(Hi) =Qi,
so part (c) of F.8.1 holds. Finally part (d) holds as Hi M = !M(LT). Thus we
have verified Hypothesis F.8.1, so we can apply the results of section F.8.
Define b, "(,etc. as in section F.8. By F.8.5.1, b ~ 3 is odd, ·so G'Y is a conjugate
of Hi and hence as DL ::; CK,
G'Y := G'Y/02(G'Y) ~Ht:= Hi/Qi= KT/Qi x DLQi/Qi
with KT/Qi an extension of L3(4) and DLQi/Qi ~ DL ~ Z2n-i.
As Di ::::) Ht and V = [V, DL], U = [U, DL]· Thus each KDL-irreducible is the
sum of n K-irreducibles J, as F4 is a splitting field for K* and n is odd. We claim
m(Ht, U) ~ 9: For if y is an involution in H+ with m([U, y]) < 9, then as m(J) ~ 9,
y+ acts on i. Then by H.4.7, either m([J, y]) ~ 4, or m(l) = 9 and m([J, y]) = 3.
So JD:= (JDL) is the sum of n ~ 3 conjugates of J, so m([JD, y]) = m([J, y])n ~ 9,
proving the claim. In particular U is not an FF-module for Ht by B.4.2.
Recall from section F.8 that Qi = CH 1 (U), there is 9b E G with "( = "(i9b,
Ai:= Z9b, D'Y := Cu-r(U), and DH 1 := Cu(U'Y/Ai).
Suppose U'Y centralizes U, so that U'Y = DT By F.8.7.7, [DHn U'Y] = l. By
F.8.7.5, [V, U'Y] # 1, so [Zi, U'Y] -=fa 1 for some l EL. If 1 # Z{ n DH 1 , then
u'Y:::; 0 21 (Cc(Zi l n DHJ ) :::; Ca(Zi) l
as Zi is a TI-set in Gin the center of TE Syb(G). Of course this contradicts the.
choice of Zf, so we conclude that 1 = Z{ n DHn and hence Z{ is isomorphic to a
subgroup of GT Therefore
4 = m2(G'Y) ~ m(Zi) = n,
so as n is odd, n = 3. As we are assuming D'Y = U'Y, [U'Y, VJ :::; Zi by F.8.7.6; so
for 1 -=fay E Zi, m([U'Y, y])::; m(Zi) = 3, contradicting m(Ht, U) ~ 9.