S22 i2. LARGER GROUPS OVER F2 IN .Cj(G, T)m 3 (H) = 1, so as n(H) = 2, and a Hall 2'-subgroup of H n Mis a nontrivial 3-
group, K/0 2 (K) ~ L 2 (4). Also 031 (H n M) ::::; 03 ' (X n M) = 0
31
(X n L) using
12.2.8, so Mi= Li(H n M) by 12.5.2.1.
Now just as in the proof of 12.5.8, Hypothesis G.2.1 is satisfied with Hi := LiH,02 (Mi), V 7 in the roles of "H, L 1 , V". Let Hi := Hi/Vi, UH := (V 7 H/, and
Q.H := 02(Hi). Then UH ::::; Z(QH) by G.2.2.1, and UH ::::; U, so that UH is
elementary abelian by 12.5.8.2. If [UH,KJ = 1 then as Vi::::; Vi, K::::; G3::::; M by
12.5.4, contrary to K f:. M. Thus as Hi= KTLi with KLiQH/QH ~ A5 x L3(2),
we conclude QH = CH(UH)·
Let Hi := Hi/QH. Now Wo := Wo(T, V) ::::; CT(V) and Na(Wo) ::::; M by
12.5.7. Hence as Hi. M, W 0 i. 02 (H) by E.3.15, so there is A:= V^9 :'::: T with
Ai. 02 (H). As A::::; Wo(T, V) ::::; CT(V) ::::; 02(Mi)::::; QH(T n K), A*::::; K*. Let
B :=An QH = CA(UH)· Then m(A/B) ::::; m2(K*) = 2. Further [UH, BJ ::::; Vi,so for u E UH, m(B/Cs(u)) ::::; m2(Vi) = 1, and hence m(A/Cs(u)) ::::; 3. Now
r(G, V) > 3 = s(G, V) by 12.5.6, sou E Na(A), and hence UH ::::; Na(A). Thus
if [UH, BJ f. 1 then V 1 = [UH, BJ ::::; A. But then 12.5.9.4 and 12.5.S.1 showA E V^01 s;; 02 (G 1 ), contrary to A f::. 02 (H). Thus UH centralizes B, so as
m(A/B) < s(G, V), A centralizes UH by E.3.6. But then [K,AJ = K centralizes
UH, contrary to our earlier observation that QH = CH(UH)·
This final contradiction completes the proof of Theorem 12.5.1.12.6. Eliminating A 8 on the permutation module
The main result of this section is Theorem 12.6.34, which eliminates the As-
subcase in case (d) ofTheorem 12.2.2.3, reducing the treatment of L/0 2 (L) ~ L 4 (2)
to case (a) where V is a 4-dimensional natural module. This leaves only one case
of Theorem 12.2.2.3 where it is possible that Cv(L) =I= 1: case (d) with L ~ A 6.That case will be treated in section 13.4 of the following chapter.
We mention that L 4 (2)/E 64 arises as LE Lj(G, T) in the non-quasithin shad-
ows G ~ nt(2), Ot 0 (2), Sps(2), and POt(3). Also such a 6-dimensional in-
ternal module appears in a suitable non-maximal member of £ 1 (G,T) in other
non-quasithin groups, such as larger orthogonal and symplectic groups, as well as
the sporadic groups J4 and F~ 4. As a result, the analysis in this case is fairly longand difficult. In particular, these shadows are not eliminated until 12.6.26.
So in section 12.6 we assume Hypothesis 12.2.3, and adopt the conventions
of Notation 12.2.5, including Z = Oi(Z(T)). In addition set Zv := Cv(L) and
V := V/Zv.
Througout this section, we assume that L ~ As and that V is the orthogonal
module for L ~ nt(2). In particular notice 02(L) = 02,z(L) = CL(V) by 1.2.1.4,
since the Schur multiplier of As is of order 2 by I.1.3.1. Further Mv = LT ~ As
or Ss = Aut(As).
We adopt the notational conventions of section B.3, and assume T preservesthe partition { {1, 2}, {3, 4}, {5, 6}, {7, 8}} of the set n of 8 points. In particular
by B.3.3, if Zv f. 1, then V is the core of the permutation module for L on n,
and Zv is generated by e,11. In that case, V :::1 M by Theorem 12.2.2.3; hence
Zv = Cv(L) :::JM, and we conclude M = Ca(Zv) as ME M. In any case Vis the
quotient of the core of the permutation module, modulo (e,11/. We can also view V
as a 6-dimensional orthogonal space for L ~ nt(2). Thus we can speak of singular