852 i2. LARGER GROUPS OVER F 2 IN .Cj(G, T)with
8i := v = u n U^9 n uh, 8 = (Un U^9 n 8) (Un uh n 8) (U^9 n uh n 8),and 8/V the sum of copies of the dual of V as an Li-module, for. each g, h E L
with V =Vi EB Vl EB Vih·
(6) u = (Vl).
PROOF. Recall <:P(U) = Vi; then (1) and (2) follow from standard arguments(cf. 23.10 in [Asc86a]). As Li is irreducible on V, either (3) holds or V:::; Z(U),
and the latter is impossible as U = (VH). By 12.8.4, Hypothesis G.2.1 is satisfied,
and we recall as in section G.2 that as U is nonabelian, the hypothesis in G.2.3 and
G.2.5 that U 1:. CT(V) = 02 (LT) is satisfied, so that (4) follows from those results.
Assume Li :::] H. Then by 12.8.5.1, tJ is the direct sum of copies of V as
a module for Li ~ Ln-i(2)'. By (2) and (3), the bilinear form ( , ) induces an
Li-equivariant isomorphism between U/Cu(V) and the dual space of V. But if
n > 3, then Vis not isomorphic to its dual as an Li-module; so we conclude n:::; 3.
Assume n = 3. Then Li~ Z3, tJ = [U, Li], and all chief factors for Li on (8nU)/V
are 2-dimensional. Therefore by (4) and G.2.5,v =: 8i = 82 = u n U^9 n uh
since [I,82]:::; V by G.2.5.5. Similarly 8 = 83, as if 8/83-/=-1, then from G.2.5.7,
Li has a I-dimensional chief factor on (Un 8)8 3 /8 3. This completes the proof of(5)..
Next
u = (VH) = (v;L1H) = W2H),
giving (6). This completes the proof of 12.8.8. DWe continue to establish analogues of results in the literature on large extraspe-cial subgroups. In Hypotheses G.10.1 and G.11.1 in Volume I, we axiomatized some
of the properties that are satisfied by Ca 0 (zo)/02(Go) acting on 02(Go)/ (zo), when
02(Go) is a large almost extraspecial 2-subgroup of a group G 0 • In 12.8.12, we ver-
ify these hypotheses in our setup, and after that we appeal to the results in sections
G.10 and G.11, particularly Theorem G.11.2.Notice for example that G.10.2 is an analogue of 3.8 in Timmesfeld [Tim78].
If G is of Lie type, with the involution centralizer Gi a maximal parabolic, the
subgroup h below corresponds to the complementary minimal parabolic. In the
theory of large extraspecial 2-subgroups, the inequality in G.10.2 typically produced
a lower bound on the 2-rank of H /CH (U). But here H is an SQTK-group over which
we have some control, so that G.10.2 serves as an upper bound on m(U), which we
then use in 12.8.12 (via an appeal to Theorem G.11.2) in order to strongly restrict
the structure of H and its action on U.
Let P be the minimal parabolic of LT acting nontrivially on Vi; notice under
part (2) of Hypothesis 12.8.1 that P =LT. Set h := (UP), W := Cu(Vi), ~nd let
g E P - H. Set E := W n W^9 , X := W^9 , and Zu := Z(U). Observe Zu:::; Was
Vi:::; u.LEMMA 12.8.9. (1) (UH) = 02 (P)U = I2 = (U, U^9 ), Cr 2 (Vi) = 02(h), and