3.2. THE FUNDAMENTAL SETUP, AND THE CASE DIVISION FOR .Cj(G, T) 579HYPOTHESIS 3.2.1 (Fundamental Setup (FSU)). G is a simple QTKE-group,
TE Syl2(G), LE .Cj(G, T) with L/02(L) quasisimple, L 0 :=(LT), M := Na(L 0 ),and Vo E Irr +(Lo, R2(LoT), T). Set V := (V[), VM := (VM), Mv := NM(V),
Mv := Mv/CMv(V), and VM := VM/CvM(Lo).
In our first lemma we apply results from section D.3 to subgroups ME M(T)
such that M is the normalizer of one of the uniqueness subgroups constructed in
chapter 1. We will also see in 3.2.3 that case (i) of 3.2.2 includes the Fundamental
Setup, as the similar notatiqn in the lemmas suggests.LEMMA 3.2.2. Assume there is M+ = 02 (M+) :::l M such that either
(i) M+ =(LT) for some LE L1(G,T) with L/0 2 (L) quasisimple, or
(ii) M+ = 02,p(M+) for some odd primep, with T irreducible on M+/0 2 ,w(M+)·
Let Vo E Irr+(M+,R2(M+T),T) and set VM := (V 0 M), V :=(Vt), and VM :=
VM/CvM(M+)· Then. (1) CM+(VM) S: 02,w(M+)·
(2) VM E 'R'2(M).
(3) VM = [VM, M+J, VM is a semisimple M+-module, and M is transitive onthe M+-homogeneous components of V M.
(4) CvM(M+) = (CVa (M+)M) = (Cv(M+)M).
(5) If Cv 0 (M+) = O;then Vo is a TI-set under M.
(6) IfCvM(M+)-/:- 0 and M = !M(M+T), then M+ = [M+,J(T)] and Vis
an FF-module for M+T.
(1) Hypothesis D.3.1 is satisfied with AutM(VM), AutM+\VM), V 0 in the rolesof "M, M+, V".
(8) VE R2(M+T) and 02(M+T) = CT(V).
(9) Assume M = !M(M+T). Then the hypothesis of Theorem 3.1.8 is satisfielf,
with M+ in the role of ''Lo", and D.3.10 applies.PROOF. By A.1.11, R 2 (M+T) S: R 2 (M). Now it is straightforward to verify
that Hypothesis D.3.2 is satisfied with M, T, M+, R 2 (M), 1, V 0 in the roles of "M,
T, M+, Q+, Q_, V". Notice that V, VM play the roles of "VT, VM" in Hypothesis
D.3.2 and lemma D.3.4. Now (1) and (7) follow from parts (2) and (1) of D.3.3.
By (7), we may apply D.3.4 to AutM(VM); then conclusions (1)-(4) and (6) of
D.3.4 imply conclusions (2)-(5) of 3.2.2. ·
Set Mo := M+T and R := 02(Mo). By D.3.4.1, 02(Mo/CM 0 (V)) = 1, so
V E R 2 (Mo) and hence R S: CT(V). By D.3.4.2, CM+(V) S: 02,w(M+), so as
M+ = 02 (M 0 ), CM 0 (V) S: R02,w(M+) and hence R = CT(V), completing the
proof of (8).
Now assume that M = !M(M+T). Then (9) follows from (8), so it remains to
prove (6); thus we assume that CvM(M+)-/:-0. Then Z 0 := Cz(M+T)-/:- 0 and Z 0 S:
Z(M 0 ). By (9) we may apply Theorem 3.1.8.3 to conclude that J(T) 1. CT(V).
From the structure of M+ in cases (i) and (ii) of the lemma, <:P(M+/02(M+)) is
the largest Mo-invariant proper subgroup of M+/0 2 (M+), so we conclude that
M+ = [M+, J(T)]0 2 (M+)· Then as M+ = 0^2 (M+), also M+ = [M+, J(T)],
completing the proof of (6), and hence of 3.2.2. D
LEMMA 3.2.3. Assume LE .Cj(G, T) with L/02(L) quasisimple, and let Lo:=