3.2. THE FUNDAMENTAL SETUP, AND THE CASE DIVISION FOR C.j(G, T) 583automophism nontrivial on the Dynkin diagram. Finally we eliminate the cases in
part (iii) of B.4.5, via an appeal to Theorem 3.1.8.2: For in these cases, q > 2 = qin the notation of B.4.5. As q > 2, case (i) of 3.1.8.2 does not hold. But V is a
TI-set under M by 3.2.7, so as q = 2, case (ii) of 3.1.8.2 does not hold either, a
contradiction. DIn our final result on the Fundamental Setup, we collect some useful properties
that hold when J(T) $ CT(V)-and hence in particular under the hypotheses of
3.2.9 where Vis not an FF-module.
Recall that Ji (T) appears in Definition B.2.2, Further n(X) appears in E.1.6,
r(G, V+) in E.3.3, and Wo(T, V+) in E.3.13.PROPOSITION 3.2.10. Assume the Fundamental Setup (3.2.1). Set V+ := V,
except in case (3.c.iii) of 3.2.6, where we take V+ := VM. Assume J(T) $ CT(V+)·
Then(1) Na(J(T)) $ M.
(2) NM(V+) controls fusion in V+.
(3) For each U $ V+, Na(U) is transitive on {v+ : U $ V+}.(4) For each U $ V+, INa(U): NM(U)I is odd.
(5) If U $ V+ with (V:a(U)) abelian, then [V+, V+J = 1 for all g E G with
u:::;v+.
(6) Suppose U $ V+ with V+ $ 02(Na(U)), and either
(a) [V+, Wo(T, V+)] = 1, or
(b) V+ is not an FF-module for AutLaT(V+)·Then [V+, v+J = 1 for each g E G with U $ v+.
(7) If Ji(T) $ CT(V+) and r(G, V+) > 1, then n(H) > 1 for each H E
H*(T,M).
(8) If J(T) $SE S2(G), then J(T) = J(S) and so Na(S) $ M.
(9) Cz(Lo) = 1 = Cv+(Lo).
PROOF. By 3.2.3, M = !M(LoT). We have CT(V+) $ CT(V) = 02(LoT) by
3.2.2.8. Hence as J(T) $ CT(V+) by hypothesis, using B.2.3.3,
J(T) = J(CT(V+)) = J(02(LoT)) ::::! LoT,
so that (1) 4olds. Notice the same argument establishes (8). Further Z(LoT) = 1
by Theorem 3.1.8.3, so (9) follows.
Observe that V+ is a TI-set under M: This holds in case (3.c.iii) of 3.2.6 as
V+ = V M is normal in M in that case, and in the remaining case V+ = V is a TI-set
under M by 3.2. 7.
Also V+ $ E := Q 1 (Z(J(T))). As J(T) is weakly closed in T, by Burnside's
Fusion Lemma A.1.35, Na(J(T)) controls fusion in E and hence in V+. Thus as
V+ is a TI-subgroup under M, (1) implies (2). Then (2) implies (3) using A.1.7.1.
Let,U $ V+ and SE Syl2(NM(U)). As J(T) $ Ca(V+) by hypothesis, we may
assume J(T) $ S. Then Na(S) $ M by (8), so SE Syl2(Na(U)), establishing (4).
Assume the hypotheses of (5), and let U $ v+. By (3), we may take g E Na(U);
then as w:a(U)) is abelian by hypothesis, [V+, V+J = 1-so that (5) is established.