1042 14. Ls(2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTYWe now derive a contradiction, hence showing that no examples satisfy the
hypotheses of this section.
By 14.6.24.4, Oa( u) EI, so that I i-0, and if Tu =Tr := T n I for some IE
I, then also Oa(u) EI by 14.6.4. By 14.6.20, IT: 02(H)I > 4, and by 14.6.22.1,
m(U /Ou(QH )) ~ 4. Thus the hypotheses of 14.6.9 are satisfied for any IE I, and
that result shows that !Tl > 211 and LT = LrTr02(LT), where Lr := 02 (L n I).
Set H 2 := OH(u). By 14.6.24.2 and 14.6.22.1, H2/0 2 (H2) ~ 83 , with H2 i. Mas
H n M = T by 14.6.23.2. By construction, To := Or(u) = Nr(UK,1) E Syb(H2),
and H 2 has nontrivial chief factors on each UK,i· Pick IE I, choosing I:= Oa(u)
if Tr = Tu for some I EI, and let I2 := 02 (H2)Tr, Ii := LrTr, and Io := (Ii, h).
Then Io EI* by 14.6.6.6. Further 02 (H 2 ) centralizes u by Coprime Action, so if
Tr = Tu, then.I2 = 02 (H2)Tu centralizes u, while Ii ::; Oa(u) by our choice of
I, so that Io :S:: Oa(u). Thus Io satisfies the hypotheses of 14.6.10.5, and hence
m((Vr^2 )) = 3 by that result. However as V = (v) ::; Z(T), from the module
structure in 14.6.22.1, v = uu 2 c, where ii2 generates Ou-K,2 (T) and c E Ou-(H).
Therefore (Vr^2 ) = (uc)[UK,2J2] is of rank 4, since Z :S:: [UK,2, 02 (H2)] by 14.6.2.This contradiction completes our analysis of the L 2 (2)-case under Hypothesis
14.2.1; namely we have now proved:
THEOREM 14.6.25. Assume Hypothesis 14.2.1. Then G is isomorphic to h,J 3 ,^3 D4(2), the Tits group^2 F4(2)' , G 2 (2)' ~ U3(3), or M12.
PROOF. We may assume that the Theorem fails, so that case (2) of Hypoth-
esis 14.3.1.2 is satisfied. By Theorem 14.3.16, U = (VG^1 ) is abelian, so that the
hypotheses of this section are satisfied. Finally, as we just saw, those hypotheses
lead to a contradiction, so the Theorem is established. D
14.7. Finishing L 3 (2) with (VG^1 ) abelian
In this section we continue to assume Hypothesis 14.5.1, but now assume that
L/02(L) ~ L3(2); that is, we treat case (1) of Hypothesis 14.3.1, so in particular
Hypothesis 13.3.l holds, with G ~ Sp 6 (2) or U4(3), and U := (VG^1 ) is abelian. Fur-
ther by 13.3.2.4, Hypothesis 12.2.3 holds, and hence so does case (1) of Hypothesis
12.8.1. Thus we can appeal to results in sections 12.8, 13.3, 14.3, and 14.5.
We will see in Theorem 14. 7. 75 that the Rudvalis group Ru is the only quasithin
example which arises under the hypothesis of this section; as far as we can tell, there
are no shadows.
We adopt Notation 12.8.2, including the T-invariant subspaces Vi of V fori = 1, 2, and the subgroups Gi := Na(Vi), Mi := NM(Vi), Li := 02 (NL(Vi)), and
Ri := 02(LiT). In particular V1 = V n Z where as usual Z := ~h(Z(T)), z is
the generator for Vi, G1 := Gi/Vi, and 1-lz consists of the members of H(L 1 T, M)
which lie in G1.
In this section, H denotes a member of 1-fz.
By 14.5.14 we may adopt Notation 14.5.16; in particular, form the coset geom-
etry r of Hypothesis F.9.1 with respect to LT and H, set b := b(r, V), choose a
geodesic
f'o, /'1, ... , /'b =: f',
define UH, U'Y, DH, D'Y, etc. as in section F.9, and set Ai:= vr where ')'1gb = ')',