i3.8. FINISHING THE TREATMENT OF A 6 96iWe claim UH has a unique maximal submodule W. Assume not; then (writing
J(UH) for the Jacobson radical of UH)
UH:= UH/J(UH) = U1 EB··· EB Us
is the sum of s > 1 four-dimensional irreducibles. Further the projection v; of
V3 on Ui is faithful for each i and centralized by Lo, so the Ui are isomorphic
natural modules. As Cu; (T*) is a point, each LiT-invariant line is contained in
a member of Irr +(H, UH), so V3 ::::; U 0 for some irreducible H-submodle U 0. But
then UH= (Vl) = Uo, contrary to s > 1. Thus the claim is established.
Define a as in case (2) of 13.8.8; thus v; ::::; 02 (LiT*) = 02 (Li) and m(V;) =m(V;) > 1.
Let B be a noncentral chief factor for H on W. We claim m(B) = 4. For
otherwise, as q(H*, UH) ::::; 2, B is of rank 6 by B.4.2 and B.4.5. Thus as v;
is a noncyclic subgroup of the unipotent radical 02 (Li) of the parabolic LiT*stabilizing a point of B and acting quadratically on B, it follows that m(V,;) = 2
and v; contains no FF*-offender on B by B.3.2.6. Therefore case (1) of 13.8.22
holds, so Ai ::::; W. As T = TK:, no member of H induces a transvection on B,
so [W, u;J >Ai and hence W i. DH by F.9.13.6. Thus we conclude from 13.8.22
that u; contains a strong FF-offender on UH. As m(V;) = 2 and u; contains a
strong FF-offender, we conclude v; = u; ~ E4. Then m(UH/DH) ::::; m(U;) =
2, so as W i. DH, m(UH/DH) ::::; 1, with m(UH/DH) = 2 in case of equality.
However u; centralizes DH by F.9.13.6 as Ai ::::; w. Therefore m(UH/DH) = 1
and m(UH/DH) = 2 = m(U;). Thus we have symmetry between ')'i and')'. In
particular as Ai ::::; W ~ UH, Vi ~ U'Y; further in view of 13.8.18.2, we may apply
13.8.11.1 to conclude u; < v;, contrary to an earlier remark. This establishes the
latest claim that m(B) = 4.
Thus we have shown that all noncentral chief factors of UH are 4-dimensional.
Then as the 1-cohomology of 4-dimensional modules is trivial by I.1.6.6, and Wis
the unique maximal submodule of fj H, all chief factors are 4-dimensional.
Observe next that no noncyclic subgroup of v; centralizes a hyperplane off) H:
For otherwise as v; is quadratic on fj H by 13.8.4.6, the quotient module fj H splits
over the submodule W by B.4.9.1, contradicting W the unique maximal submodule
of UH. So as v; lies in the uni potent radical 02 (Li) of the stabilizer of a line in
the natural module for L 4 (2), it follows that m(U;) ::::; m(V,;) ::::; 3.
Now let J denote any member of Irr+ ( H*, W), so that in particular J is 4-
dimensional. Applying the dual of B.4.9.1, we conclude similarly that no noncyclic
subgroup of v; acts as a group of transvections with a fixed center on J.
Suppose next that Ai ::::; J. Then CH(Ai) is the maximal parabolic fixing Ai.
Then as H = Gi, V 7 :::'.! Na(Ai), so v; = 02(CH(Ai)) as Na(Ai) is irreducible
on 02(CH(Ai)). This is impossible as v;::::; 02(Li) where LiT stabilizes a line
of J.
Therefore Ai i. I, so [InDH, U 7 ]::::; JnAi = 1; then by 13.7.3.7 1 [InDH, V'Y]::::;
Ai n I = 1. In particular I i. DH.
Suppose next that m(u;) = 1. We saw v'Y centralizes DH n I, which is a
hyperplane of I by 13.8.23.1. Then as v,; ::::; 02 (Li) and LiT is the parabolic
stabilizing a line in J, we conclude m(V,;) ::::; 2, and hence m(V,;) = 2 as m(V;) > 1