796 12. LARGER GROUPS OVER F2 IN .Cj(G, T)one of the conclusions of Theorem 12.2.2 holds-namely (a), (b ), the subcase of
(d) with Sp 4 (2)' ~ A 6 , or (f). Thus Theorem 12.2.2 holds in the first four cases of
3.2.8. In the remaining cases of 3.2.8, one of the conclusions of part (3) of Theorem
12.2.2 holds; notice that 3.2.8.10 correponds to the subcase of case (d) of conclusion
(3) of 12.2.2 where L ~ L 4 (2) ~ A 8 • So the proof is complete. D
Thus in the remainder of this section, and in many subsequent sections, we will
assume:HYPOTHESIS 12.2.3. Hypothesis 12.2.1 holds, and G is not one of the groups inconclusions (1) and (2) of Theorem 12.2.2. Thus conclusion (3) of Theorem 12.2.2
holds. Set M := Na(L), and let VE Irr +(L, R 2 (LT), T).
Since Hypothesis 12.2.3 implies that conclusion (3) of Theorem 12.2.2 holds,
the remainder of our treatment of the Fundamental Setup is devoted to the groups
and modules listed there.Observe also that Hypothesis 12.2.3 imposes constraints on all members of
Cj(G,T):
REMARK 12.2.4. Assume Hypothesis 12.2.3. Then for any K E Cj ( G, T) with
K/0 2 (K) quasisimple, Hypothesis 12.2.3 holds for K in the role of "L". Thus
K is described in conclusion (3) of Theorem 12.2.2, T normalizes K, there exists
VK E Irr +(K, Rz(KT), T), and any such VK is described in conclusion (3) of
Theorem 12.2.2.Indeed observe that any KT-submodule of R 2 (KT) which is irreducible module
K-fixed points must contain such a VK, and hence must itself be of the form VK·However rather than introducing further notation for KT, we will continue to use
the existing notation of Irr +(K, R 2 (KT), T).
Usually when we assume Hypothesis 12.2.3, we adopt the following notational
conventions:
NOTATION 12.2.5. (1) Z := fli(Z(T)) and Q := 02 (LT).
(2) Mv := NM(V) = Na(V) and Mv := Mv/CMv(V).
(3) For v E V#, Gv := Ca(v), Mv := CM(v), Lv := 02 (CL(v)), and Tv .-
CT(v). We have the properties:
(a) Lv :::;l Mv.(b) Conjugating in L, we may choose v so that Tv E Syl2(Mv)·
(c) Mv E He.
(d) C(Gv, Q) ~ Mv.(e) Mv ~ Mv.
(4) For z E vnz#, set Gz := Gz/(z).
PROOF. We establish the properties claimed in part (3): Part (a) follows since
L :::;! M; (b) follows since T E Sylz(G), (c) follows from 1.1.3.2; (d) follows as
C(G, Q) ~ M by 1.4.1.1; (e) is a special case of 12.2.6, established in the next
subsection. D
12.2.1. Preliminary'results under Hypothesis 12.2.3. Recall that we are
assuming Hypothesis 12.2.3, so that in particular Theorem 12.2.2.3 holds. Our first
result follows from Theorem 12.2.2.3 and 3.1.4.1: