580 3. DETERMINING THE CASES FOR LE L:.j(G, T)Irr +(L 0 , R 2 (L 0 T)) there. exists Vo E Irr +(Lo, R2(LoT), T) with Vo/CvJLo) Lo-
isomorphic to I/Cr(L 0 ). In particular L and V := (V;,T/ satisfy the Fundamental
Setup {3.2.1).PROOF. By 1.2}.3, M = !M(L 0 T). By A.1.42.2, there exists a member Vo
of Irr+(L 0 ,R2(LoT),T) with Vo/Cv 0 (Lo) isomorphic as Lo-module to I/Cr(Lo).
Hence the lemma holds. DREMARK 3.2.4. Given LE Cj(G, T) with L/0 2 (L) quasisimple, lemma 3.2.3
shows that we can choose V so that L and V satif;lfy the Fundamental Setup.
Then by 3.2.2.7, we may apply the results of section D.3 to analyze V, VM, and
AutL 0 (VM)· By 3.2.2, we may also appeal to Theorem 3.1.8, and in view of 3.2.2.4,3.2.2.6 supplies extra information when Cv(L) =/= 0.
In the next few lemmas, we determine the list of modules V and VM that can
arise in the Fundamental Setup for the various possible L E Cj ( G, T). The first
result 3.2.5 below gives us a qualitative description of what goes on in the case
L = L 0 , including a fairly complete description of the case where Vo < V. Then3.2.8 gives more detailed information when L =Lo but Vo = V.
Recall that VM := (v;,M) and that V plays the role of "Vr" played in lemma
D.3.4. Also recall that in the FSU, Mv denotes NM(V)/CM(V).
THEOREM 3.2.5. Assume the Fundamental Setup {3.2.1}, with L =Lo. Then
q(Lf', V):::;: 2 2 q(AutM(VM), VM), and one of the following holds:{1} V 0 = V = VM; that is, V 0 :::;! M.
{2} V 0 = V :::;! T, CvJL) = 0, and Vis a TI-set under M.
{3} L 2:! SLa(2n) or Sp4(2n) for some n, A5, L4(2), or Ls(2); CVo(L) = 0 and
either Vo is a natural module for L or V 0 is a 4-dimensional module for L 2:! A1;
and VM = V =Vo EB v; with t ET-Nr(Vo), and v; not F2L-isomorphic to Vo.
PROOF. As discussed in Remark 3.2.4, we may apply 3.2.2, Theorem 3.1.8, andresults in section D.3. Recall that in our setup, V 0 and V play the roles of "V" and
"Vr" in Hypothesis D.3.2 and lemma D.3.4.Set q := q(Lf', V) and q := q(AutM(VM), VM)· As L = Lo by hypothesis,
conclusion (1) of Theorem 3.1.8 gives q:::;: 2.Next we will show that q:::;: 2 by an appeal to Theorem 3.1.6. Set R := Cr(VM ),
so that RE Syl2(CM(VM )). We first verify that for any HE H*(T, M), Hypothesis
3.1.5 is satisfied with M 0 := NM(R) and VM in the role of "V": First as VM :::;! M,
hypothesis (II) of 3.1.5 is satisfied. By a Frattini Argument, M = CM(VM )Mo,
so AutM(VM) 2:! AutM 0 (VM), and hence as VM E R2(M) by 3.2.2.2, also VM E
R2(Mo). As R :::;! Mo, R:::;: 02(Mo). "As VM E R2(Mo), 02(Mo):::;: CM(VM), so as
R is Sylow in CM(VM), R = 02(M 0 ). This completes the verification of Hypothesis
3.1.5.
Next V :::;: VM, so R :::;: Cr(V), while Cr(V) :::;! LT by 3.2.2.8. Thus R =
Cr(V)nCM(VM) :::;! LT, so as M = !M(LT), M = !M(Mo). Therefore conclusion
(1) of Theorem 3.1.6 is not satisfied, so one of conclusions (2) or (3) holds, and in
either case, q :::;: 2 as desired.
We have shown that q :::;: 2 2 q, so it remains to show that one of conclusions
(1)-(3) holds. Suppose first that CvJL) =/= 0. Then by 3.2.2.6, L = [L, J(T)], so
that (in thelanguageofDefinitionB.1.3) AutL(VM):::;: J(AutM(VM), VM) byB.2.7.
Thus we have the hypotheses for D.3.20, which gives conclusion (1). Therefore we