8.3. ELIMINATING La(2) 12 ON 9 723L;/02(L;) ~ A5 that the A5-module V/Vz does not arise in A.3.14.^3 That is, (1)
holds.
By 8.2.12.1, Eis of rank 12. Therefore by H.12.1, E is irreducible and
EndK;(E) ~ F4, so Z(K;) = Ca.(K;). Finally there is t E TnL inducing an
outer automorphism on Lz/02(Lz) and hence also onK;, so as IAut(K;): K;I = 2,C = TKzCo(Kz) = TKz with Kz of index 2 in C. Therefore CM(z) = LzT with
ITI = 221 =IL n Tl. Then as M = LCM(z), L = M, so (2) holds. D
As a corollary we get:
THEOREM 8.2.14. G ~ J4.
PROOF. By 8.2.12, E = F*(C) ~ D~, and by 8.2.13, C/E ~ Aut(M 22 ). Also
zL n Vz -=/= { z}, so z is not weakly closed in E with respect to G. These are thehypotheses of Aschbacher-Segev [AS91], so we conclude from the main theorem
of that paper that G ~ J4. We mention that their work uses the graph-theoretic
methods used elsewhere in this work to establish recognition theorems. D8.3. Eliminating L 3 (2) 12 on
In this final section of chapter 8, we treat the exceptional case of L 3 (2) ( 2 on its
tensor product module, which we have been postponing since the previous chapter.
We prove:
THEOREM 8.3.1. The case L 0 ~ L3(2) x L3(2) on its 9-dimensional tensor
product module cannot ari:se.We begin by defining notation: Let Li := L, L2 := Lt, Lo := L1L2, Vi E
Irr +(L1, V) with Vi NT(L)-invariant, and V2 := Vf, so that Vis the tensor product
of Vi and Vi as an L 0 -module. Thus we can appeal to subsection H.4.4 of chapter
Hof Volume I.
Let Vi,m be the NT(L)-invariant m-dimensional subspace of v;, and adopt
the following notation for the unipotent radicals of the corresponding parabolic
subgroups: R;, := CTnL;0 2 (LoT) (Vi,2), and Si := CrnLi0 2 (LoT) (Vi/Vi,1). Let R :=
R1R2, S := S1S2, and as usual set Q := 02(LoT). Notice To := RS is Sylow in
L 0 Q, and of index 2 in T. Let Wj := Wj(T, V) for j = 0, 1.
From 3.2.6.2, we have V = VM, so M = Mv.
LEMMA 8.3.2. s(G, V) = 3.
PROOF. This follows from 7.3.2 and Table 7.2.1.Recall from 7.3.3 and Table 7.2.1 that w 2: 1; indeed we can show:
LEMMA 8.3.3. Either(1) W 1 centralizes V, so that w > 1; or
(2) W1 =fl and W1(S, V) = W1(Q, V).
D
PROOF. Suppose A::::; vg n T with m(Vg /A)::::; 1, but Ai= 1. By 8.3.2, s = 3,
so that A E A 2 (M, V) by E.3.10. Then by H.4.11.2, A ::::; R. So if W1 does not
centralize V, W 1 = R since NM(R) is irreducible on R. Similarly Rn S contains
no members of A 2 (M, V), so W1(S, V) = W1(Q, V). D
(^3) We just eliminated the shadows of U5(2) and 002, where G/E e! U4(2), Sp5(2) are not
SQTK-groups.