i4.7. FINISHING L 3 (2) WITH (VG1) ABELIAN 1057A5 wr Z2 extended by an involution inducing a field automorphism on both K*
and Kt. In either case no element of H induces a transvection on U. Therefore
D"f < U"f by 14.5.18.1, so we may adopt Notation 14.7.1.
Let J be a maximal H-submodule, and set W := [J / J, so that W is H-irreducible. Let Vw denote the image of V in W. By 14.7.2.1 applied to Li in
the role of "Y", Vw is isomorphic to Vas an Li-module. Next as His irreducible
on W, either W is the tensor product Wi ® W 2 of irreducibles Wi for Ki := K
and K2 := Kt, or W = Wi EB W2 with Wi := [W, Ki] a Ki-irreducible. But in
the latter case there is no BT-invariant line Vw of W with Vw = [Vw, Li]. ThusW = Wi ® W2, and a similar argument shows that each Wi is the L 2 (4)-module,
so that W is the orthogonal module for K 0 ~ nt(4), and Vw is the T-invariant
singular F 4-point. Let To := T n Ko. By (*), H* = K 0 T*, so as K 0 is faithful on
w, so is H*; then as u; E Q(H*' U), u; E Q(H*' W). If a* is an involution in H*
then either Cu-(a*) = [U,a*] or a* induces an F 4 -transvection. Thus as u; is qua-dratic on U, Cu-(U;) = Cu-(a) for each a Eu;# which is not an F 4 -transvection,
and in particular for each a* E K 0. Then calculating in the orthogonal module, we
conclude that one of the following holds:
(i) u; = (t), t an F4-transvection, and [W, t] is a nonsingular F4-point of W.
(ii) u; is a 4-group with [W, U"f] = Cw(U"f) of rank 4.
(iii) u; = (t)F, where F := CT 0 (t) ~ E 4 , and [W,U'Y] = Cw(U"f) is of
rank 4.
Suppose case (iii) holds. Then 3 = m(U;) ~ m(U / D) by choice of 'Yin Notation
14.7.1, while by F.9.13.6, [.D, U"f] ::::; Ai with m(Ai) = 1. But then the image Dw
of D has corank at most 3 in W, so Dw is not Cw(U"f), and we compute in
the orth.ogonal module W that [Dw, U"f] has rank at least 2. This contradiction
eliminates case (iii).
Thus case (i) or (ii) holds. Then as u; E Q(H*,U) with m(W/Cw(U"f)) =
2m(U;), we conclude that W is the unique noncentral H-chief factor on U, and
W = [U, K 0 ]. Turther as Li ::S: Ko with V = [V, Li], U = (VH) = W. By 14.5.18.2,
m(U;) = m(U/D), so we have symmetry between 'Yi and 'Y (cf. Remark 14.7.17),
and u; acts faithfully as a group of F2-transvections on D with center Ai. This
eliminates case (ii), for there D has corank 2 in [J = W, while as u; contains a free
involution, U"f does not induce a 4-group of F 2 -transvections with fixed center on
any subspace of corank 2. It also shows Ai ::S: U, and hence by symmetry, Vi ::::; U"f.
Thus case (i) holds. Recall that under Notation 14.7.1, we choose a and h so
that Ua::::; Ri, and as in Remark 14.7.17, we also have symmetry between 'Yl and
a. Then U~ = (t), where t E T* induces an F4-transvection on [J = W, and
[U, t] is a nonsingular F 4 -point. We also saw that m(U/D) = 1 and that t* induces
an F2-transvection on the F2-hyperplane D of U with [D, t] = .Aq.
To complete the proof, we will define subgroups Y, Vy to which we apply
14.7.31.2, to construct a 2-local I, which we then use to derive a contradiction. We
saw that Vis the T-invariant singular F4-point in U containing V 2 , and H = G1,
so Ca(Vi) = CH(Vi) ::S: NH(V) ::S: M.
Set Vy:= V1Aq ~ E4. As His irreducible on U, [Aq, QH] =Vi by 14.5.21.1,