794 12. LARGER GROUPS OVER F2 IN .C'f (G, T)totally singular subspace of Cv(X), dim(Cv(X)) 2 2dim(U). Thus
dim(V/U) 2 dim(V/Cv(X)) + dim(Cv(X))/2
= 2n - dim(Cv(X))/2 2 2n - (2n - 4)/2 = n + 2,
establishing (3).LEMMA 12.1.10. W 2 (T, V) centralizes V, so that w > 2.
D
PROOF. Assume that W 2 (T, V) i Cr(V). Then w = 2 by 12.1.8, so there is a
w-offender A:= V9 n M:::; T with m(V9 /A) = 2 and A =I-1. Let U := Nv(V9);then m(V/U) 2 2 as w = 2. By 12.1.9.3, m(A) 2 n. Then by E.3.32,
f' n-l,A (V) = f' n-l,A (V) :::; U < V. ( *)
Suppose first that n = 4. Then m(A) = 4 = m 2 (M), so by lemma H.9.3.3, we
may take A to be one of the subgroups there denoted by Ai for 0 :::; i :::; 2. Set
Bi := Ai n L. By (*) and H.9.3.4, i =I-0. Then we conclude from the last two
parts ofH.9.3 that f' 2 ,A:(V) is of rank 6. As m(V/U) 2 2, U = f' 2 ,A:(V) = f' 3 ,A:(V),
whereas Cv(a) i f' 2 A:(V) for a E A 1 - B 1 , or for a a transvection in B2·
This contradicti~n reduces us to the case n = 5. Then by lemma H.9.2.3, we
may take A :::; Ao in the notation of that result. Now lemma H.9.2.5 contradicts
(*), completing the proof. D
We are now in a position to establish a contradiction. Pick HE 'H*(T, M). By
12.1.9.2, n(H) = 2. However by 12.1.10, w 2 3, while by 12.1.9.3, r(G, V) 2 6.
Thus Hi M with n(H) < min{w, r(G, V)}, contrary to.E.3.35.1.
This contradiction shows:
THEOREM 12.1.11. Assume G is a simple QTKE-group, T E Syl 2 (G), and
LE Cj(G,T) with L/02(L) ~ £4(2) or £ 5 (2). Let M := Na(L). Then there is
no V E R2(M) such that M/CM(V)) ~ Aut(L/0 2 (L)) and V is the sum of the
natural module and its dual for L/0 2 (L).
By Theorems 12.1.11, 3.2.5, 3.2.8, and 3.2.9, the subcase of the FundamentalSetup with L/02(£) ~ £4(2) or £ 5 (2) is reduced to the cases (i.e. cases (9), (10),
and (11) of 3.2.8) with V/Cv(L) a natural module or its exterior square. These
cases will be treated along with the other cases where L/0 2 (£) is defined over F 2 ;
in particular they are completed in section 12.6, and in the final three sections of
this chapter.
12.2. Groups over F 2 , and the case V a TI-set in G
We now begin a fairly unified treatment of those simple QTKE-groups G for
which there exists L E .Cj(G, T) such that the section L/0 2 (£) has not yet been
eliminated from the list of cases in section 3.2. Thus in section 12.2, and indeed in
many subsequent sections, we assume the following hypothesis:
HYPOTHESIS 12.2.1. G is a simple QTKE-group, T E Syl2(G), a'nd L E.Cj(G, T) with L/02(L) quasisimple.
We begin with a Theorem which summarizes much of what we have accom-
plished up to this point: