CHAPTER 7
Eliminating cases corresponding to no shadow
Recall we wish to prove:THEOREM 7.0.1. Assume the Fundamental Setup (3.2.1). Then one of the
following holds:
(1) Vis an FF-module for NaL(V)(AutL 0 (V)).
(2) Vis the cocode module for M/V ~ M 24 and G ~ J4.
(3) Vis the f'l"4_(2n)-module for AutL 0 (V) ~ L2(2^2 n).
(4) Conclusion (3) of 3.2.6 is satisfied. In particular L <Lo and L/0 2 (L) ~
L2(2n), Sz(2n), or L3(2).Recall also that in Part 3, we concentrate on the cases in the FSU not appearing
in cases (1), (3), or (4) of Theorem 7.0.l; so we assume the following hypothesis:
HYPOTHESIS 7.0.2. (1) The Fundamental Setup (3.2.1) holds. In particular
LE .Cj(G, T) with L/02(L) quasisimple, Lo:= (L'{;), and M := Na(L 0 ).
(2) Vis not an FF-module for NaL(V)(AutL 0 (V)).
(3) Case (3) of 3.2.6 does not hold.
(4) Vis not.the orthogonal module for AutL 0 (V) ~ n4(2n).
Part (1) of Hypothesis 7.0.2 has various consequences including the following:
As L E .C*(G, T), by 1.2.7.3 LoT is a uniqueness subgroup with M =!M(LoT).
Furthermore by 3.2.2.8, our module V for L 0 T is 2-reduced, and we have various
other properties including Q := 02 (L 0 T) = CT(V), V :SJ T, and M = !M(Na(Q)),
so that C(G, Q) :::; M, as in 1.4.1.
By part (2) of Hypothesis 7.0.2 and Remark B.2.8, J(T) :::; Ca(V), so Q con-
tains J(T). By 3.2.10, a number of useful properties follow from this fact; for
example, Na(J(T)) :::; M, so that J(T) :::; S :::; T implies Na(S) :::; M. Further
there are restrictions on the subgroups H E H*(T, M): By 3.1.8.3, H centralizes
Z := D1(Z(T)) and Cv(Lo) = 1.
Finally by part (3) of Hypothesis 7.0.2 and 3.2.7, V is a TI-set under M. It
follows that H n M:::; CM(Z):::; Na(V) = Mv.
In this chapter we begin the anaysis of groups satisfying Hypothesis 7.0.2. In
the first section, we list the cases that can arise. The last of these cases seems
difficult to treat using only the methods of this chapter, so in the third section we
also add Hypothesis 7.3.1, which excludes that case; the case is treated in the final
chapter of part 3. Also the penultimate case and the case where Lo/02(L 0 ) ~ L3(2)
and m(V) = 6 cause difficulties, requiring extra analysis; these cases are treated in
the last sections of this chapter and the next chapter.
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