CHAPTER 9
Eliminating nt(2n) on its orthogonal module
The results in chapters 7 and 8 almost suffice to establish Theorem 7.0.1, ourmain result on pairs L, V in the Fundamental Setup (3.2.1) where V is not an
FF-module. The only case left to treat is the case where L 0 /0 2 (L 0 ) ~ L 2 (2n) x
L2(2n) ~ nt(2n) with n > 1, and Vis the orthogonal module for Lo/02(L 0 ).
The standard weak closure arguments that handle most of the pairs in chapters
7 and 8 are not so effective in this case. Difficulties are already apparent from the
parameters in Table 7.2.1: For example if T contains an orthogonal transvection
a, then m(M, V) = n, so that if n = 2 we cannot immediately apply Theorem
E.6.3 to obtain r(G, V) ~ m(M, V) as in 7.3.2. We are able to circumvent this
difficulty in Lemma 9.2.3 below. There are more serious problems, however: First,
a(M, V) = n =.s(G, V), so 7.3.3 is ineffective. Second, Lis not normal in M, so
we can't appeal to 7.4.1 to get an effective lower bound on r. Thus we will instead
use the fact that G is a QTKE-group to restrict various 2-locals, in order to show
that r is large and n(H) is small for each HE H*(T, M). Then weak closure will
become effective.
9.1. PreliminariesWe begin by establishing some notation and a few properties of M.
Let F := F 2 n and regard V as a 4-dimensional orthogonal space Vp over F.
As usual, let Q := 02 (L 0 T). Notice that we are in case (1) of 3.2.6, and in that
case V=VM ::::! M, so Mv =M.
LEMMA 9.1.l. Lo= QP' (M) for each prime divisor p of 22 n'_ l.
PROOF. This follows from 1.2.2.a. DLEMMA 9.1.2. (1) M := M/Co(V) is a subgroup of NoL(V)(Lo) = NrL(VF)(Lo),
which is the product of Lo with the F-scalar maps, extended by (!,a) ~ Z2 x Zn,
where a induces an F-transvection on VF normalizing T, with J,o- = Lt, and f
generates the group of field automorphisms (simultaneously) on L and D.
(2) There are elements in T - NT ( L) of the form a f o with f o E 02 ( (!)).
(3) Lo has two orbits on F-points of V, consisting of the singular and nonsin-
gular F -points.
(4) VN := [V,a] is a nonsingular F-point, and setting LN := 02 (NL 0 (VN)),
NLodVN) = LNQ with LN ~ L2(2n) and [V,LN] = Cv(CT) = v.zt an indecompos-
able 3n-dimensional LN-module, with Cv(CT)/VN the natural LN-module.
(5) Let V1 denote the singular F-point stabilized by T. Then NL 0 T(V1) is a
Borel subgroup of L 0 T, and is transitive on Vl.
PROOF. This is straightforward. D729