3.2. THE FUNDAMENTAL SETUP, AND THE CASE DIVISION FOR .Cj(G, T) 581may assume that Cv 0 (L) = 0. Then by 3.2.2.5, Vo is a TI-set under M. If V 0 = V
then conclusion (2) holds, so we may assume that V 0 < V. As q ~ 2 2: q, thehypotheses of Theorem D.3.10 are satisfied; therefore as we have reduced to the
case where Vo < V, conclusion (2) of Theorem D.3.10 holds. But this is precisely
conclusion (3) of Theorem 3.2.5, so the proof is complete. DThe notation Q(X, W) appears in Definition D.2.1.THEOREM 3.2.6. Assume the Fundamental Setup (3.2.1) with L < L 0. Set
M* := M/CM(VM ), U := [VM, L], and let t ET - NT(L). Then q(L 0 f', V) ~ 2 2:q(M*,VM), and one of the following holds:
(1) L* ~ L2(2n) and Vo= V = VM is the nt(2n)-module for L 0.
(2) L* ~ L3(2) and Vo = V = VM is the tensor product of natural modules for
L* and L*t.
(3) Each of the following holds:
(a) VM = U EB f)t, where U = [VM,L] ~ CvM(Lt).
(b) Each A E Q*(M*, VM) acts on U, so q(AutLaT(U), U) ~ 2.(c) One of the following holds:
(i) U =Vo and V = VM·
(ii) AutM(L*) ~ Aut(L3(2)), V = VM, U =Vo EB V 08 for s E NT(L)-
L02(LNT(L)), and m(V 0 ) = 3.(iii) L* ~ L 3 (2), U is the sum of four isomorphic natural modules for
L*, and 02 (CM·(L 0 )) ~ Z5 or E25·
PROOF. Proceeding as in the proof of Theorem 3.2.5, and recalling the dis-
cussion in Remark 3.2.4, we verify Hypothesis 3.1.5 for Mo := NM(R) where
R := CT(VM), and apply Theorems 3.1.6 and 3.1.8 as before to conclude
q(Lof', V) ~ 2 2: q(M*, VM)·Recall from the remark before that result that we may reduce case (3) to case (1)
by a new choice of V. If V < VM, then conclusion (2) of D.3.21 holds, so that
conclusion (3) of 3.2.6 holds, with case (iii) of part ( c) of (3) satisfied.So we may suppose instead that V = VM, as in conclusion (1) ofD.3.21. Assume
first that Vo < V. In particular we have the hypotheses of D.3.6, and conclusions
(1) and (2) of that result give parts (a) ·and (b) of conclusion (3) of 3.2.6, while the
two alternatives in part (3) of D.3.6 are cases (i) and (ii) of part (c) of conclusion
(3) of 3.2.6.Thus the Theorem holds when V 0 < V, so assume instead that V 0 = V. Then
we have the hypotheses of D.3.7, and its conclusions (1) and (2) give the corre-
sponding conclusions of 3.2.6. The proof is complete. D
We often need to know that Vis a TI-set under M. The previous two results
say that this is almost always the case:
LEMMA 3.2.7. Assume the Fundamental Setup (3.2.1). Then either(1) V is a TI-set under M, or
(2) L ~ L 3 (2), L < L 0 , and subcase (3.c.iii) of Theorem 3.2.6 holds.
PROOF. Suppose V is not a TI-set under M. Then in particular V is not
normal in M, so that V < V M. Therefore L < Lo, since if L = Lo then either