i3.3. STARTING MID-SIZED GROUPS OVER F2, AND ELIMINATING U 3 (3) 889PROOF. Recall Vis a TI-set in M by 12.2.6, so Hypothesis E.6.1 is satisfied,
and for 1 # U ::; V, CM(U) ::; Mv. By parts (4)-(6) of B.4.6, m(Mv, V) > 1,so CM(W) = CM(V) for each hyperplane W of V. Further the hyperplanes of V
are of the form VJ_ for v E v#, so as L is transitive on V#' L is transitive on
hyperplanes. Hence each hyperplane is invariant under a Sylow 2-subgroup of LT,
so that r(G, V) > 1 by E.6.13. Hence (1) is established.
Next we establish some preliminary results, phrased in terms of the usual ge-ometry of points and lines on V: From section 5 in [Asc87], we can identify the
points and lines of the generalized hexagon of Go := NaL(V)(L) ~ G 2 (2) with
the points and doubly singular lines of V (i.e., totally isotropic as well as singu-
lar in the Dickson trilinear form; see p. 194 of [Asc87]). By 5.1 in [Asc87],
Go is transitive on nondegenerate lines of V, and each such line l is generatedby a pair u, v of points opposite (i.e., at maximal distance) in the hexagon. Now
Na 0 (l) =Hix fI2, where Hi:= Ca 0 (l) = Ca 0 (u) n Ca 0 (v) ~ 83 by F.4.5.5, and
ff 2 := Ca 0 (Hi) ~ 83 acts faithfully on l. Further fiifI 2 acts faithfully on the4-space l.l, with l.l = [l.l, 02 (Hi)]. Now Nillv(l) is of index 1 or 2 in Na 0 (l) in the
cases Mv =Go or L, respectively. In particular Q := 02 (LT) is of index at most
2 in TH:= CT(l), so J(TH) = J(Q) in view of B.4.6.13. Further Q = 02 (KiTH),
where Ki := 02 (Hi), and K 2 := 02 (H 2 ) induces Z 3 on l. Set H := Ca(l), so that
TH E 8yb(H n M). As Na(Q) :SM= !M(LT), C(H, Q) :SH n M =: MH. In
particular as J(TH) = J(Q), NT(TH)::; MH, so that THE 8yl 2 (H). It also follows
as Ki::; MH that Q = 02(MH) = 02(NH(Q)). Thus Hypothesis C.2.3 is satisfied
with Q in the role of "R".
We are now ready to establish our main preliminary result: we claim that
H = Ca(l) ::; M. So we assume that H 1:. M, and derive a contradiction. Observe
first that as l contains 2-central involutions, H E He by 1.1.4.3. Next Q is Sylow
in 02 ,p(H)Q by C.2.6.1, and as M = !M(LT),
NH(Wo(Q, V)) '.S MH 2 CH(Ci(Q, V)).
Hence as n(0 2 ,F(H)) = 1 by E.1.13, 02 ,p(H) ::; M by (1) and E.3.19. On the other
hand, if 02,F*(H) :S MH, then 02(H) = Q by A.4.4.1; thus H :S Na(Q) :SM,
contrary to our assumption.
This contradiction shows that there is K E C(H) with K/0 2 (K) quasisim-
ple, and K 1:. M. By 1.1.3.1, K E He. Suppose first that Q 1:. NH(K). Then
C.2.4.2 shows that Q n K E 8yl2(K), and as K 1:. M, C.2.4.1 then shows that
K is a xo-block. In particular m 3 (K) = 1 and hence m 3 ((KQ)) = 2. But then
m3(K2(KQ)) 2 3, contrary to Na(l) an SQTK-group.
Therefore Q::; Na(K), so that K is described in C.2.7.3. Notice if case (g) of
C.2.7.3 occurs with n even, then we are in one of cases (1)-(4) of C.1.34, in which
Z(02(K)) is the sum of at most two natural modules for K/02(K) ~ 8L 3 (2n);this case is ruled out by A.3.19 as K2 1:. K. The remaining cases of C.2.7.3 where
m 3 (K) = 2 are eliminated by A.3.18 as K 2 1:. K. Thus m 3 (K) = 1. Also K is not
an A5-block as MH contains the Sylow 2-group TH of H. Thus inspecting C.2.7.3,
one of the following holds:
(i) K is an L 2 (2m)-block, Q is Sylow in KQ, and MK := MH n K is a Borel
subgroup of K.
(ii) K/02(K) ~ L3(2n), n odd, MK is a maximal parabolic of K, and K is
described in C.1.34.