4.4. CONTROLLING SUITABLE ODD LOCALS 619K2/02(K2), contrary to CR 2 (X)/CR 2 (K2X) the natural module for K 2 /0 2 (K 2 ).
This contradiction completes the proof of Theorem 4.3.2.THEOREM 4.3.17. If S::::; T with Sn LE Syl2(L), then Na(S) ::::; M.
PROOF. By Theorem 4.3.2, M = !M(L), so the assertion follows from 4.3.1.
D4.4. Controlling suitable odd locals
In this section, we apply Theorem 4.2.13 to force the normalizers of suitable
subgroups of odd order to lie in M. The main results are Theorem 4.4.3 and its
corollary Theorem 4.4.14.
During most of this section, we assume:HYPOTHESIS 4.4.1. {1) Hypothesis 4.2.1 holds. Set M+ := (LT) and R+ :=
02(M+T) = CT(M+/02(M+)).
(2) 1 f-B::::; CM(M+/02(M+)), with B abelian of odd order and BT+= T+B
for some T+::::; T with LT= LT+.{3) 1 f-VB= [VB, M+]::::; CM(B) with VB an M+T-submodule of D 1 (Z(R+)).
REMARK 4.4.2. Observe that if L ::::;! M, then it is unnecessary to assume the
existence of T+. For example, we could then take T+ = 1. Thus if Hypothesis 4.2.1
holds with L ::::;! M and V E R2(LT) with [V, L] f-1, then appealing to 1.4.1.4,
Hypothesis 4.4.1 is satisfied for each nontrivial abelian subgroup B of CM(V) of
odd order with Vin the role of "VE".
In this section we prove:
THEOREM 4.4.3. Assume Hypothesis 4.4.1. Then either
(1) Na(B)::::; M; or
{2) L ::::;! M, L/02(L) is isomorphic to L 2 (2n), L3(2), L4(2), A 6 , A 7 , A5, or
U3(3), and one of the following holds:
{i) VB is an FF-module for LT/CLT(VB)· Further:
(a) If L/02(L) 9:! Ln(2), then either VB is the sum of one or more
isomorphic natural modules for L/0 2 (L), or VB is the 6-dimensional orthogonalmodule for L/02(L) 9:! L4(2). ·
{b) If L/02(L) 9:! A5, then for each z E CvB (TnL)#, VB i 02(Ca(z)).
{c) If L/02(L) 9:! U3(3) and m(VB). = 6, then Ca(Vi) i M, for Vi
the (T n L)-invariantsubspace of VB of rank 3.{ii) L/02(L) 9:! L2(2^2 n), and VB is the D4(2n)-module.
{iii) L/02(L) 9:! L3(2), and VB is the core of a 7-dimensional permutation
module for L/02(L). ·
Set GB := Na(B), MB := NM(B), LB := CM+(B)^00 , and TB := NT+(B).
Making a new choice of T+ if necessary, we may assume TB E Syl 2 (MB)· As G is
simple, GB< G, so GB is a quasithin JC-group.
Before working with a counterexample to Theorem 4.4.3, we first prove two
preliminary lemmas which assume only parts (1) and (2) of Hypothesis 4.4.1.