CHAPTER 15
The case £r(G, T) = 0
In this chapter, we complete the treatment of the case £1(G, T) empty. Since
the previous chapter completed the analysis of the case £j(G, T) nonempty, this
chapter will complete the proof of our Main Theorem.
Initially we assume Hypothesis 14.1.5, introduced at the start of the previous
chapter, with M := Mf. Recall that V(M) is defined just before 14.1.2: as men-
tioned in section A.5, in this chapter we are deviating from our usual meaning
of V(M) in definition A.4.7, instead using the meaning in notation A.5.1, namely
V(M) := (ZM). In the first two sections of this chapter, we reduce to the case
where Mand V := V(M) satisfy m(V) = 4 and M/0 2 (M) ~ ot(V). We treat
that final difficult case in the third section. The fourth section then treats the
remaining subcase of the case £1( G, T) empty when Hypothesis 14.1.5 is not sat-
isfied; this subcase quickly reduces to the situation £( G, T) empty, or equivalently
each member of H(T) is solvable.
15.1. Initial reductions when £r(G, T) is empty
In this section, and indeed until the final section of this chapter, we assume
Hypothesis 14.1.5. This Hypothesis isolates the most important subcase of the case
£, f ( G, T) empty, and was already introduced at the beginning of the previous chap-
ter. Recall Hypothesis 14.1.5 includes the assumption that IM(T)I > 1, which is
appropriate in view of Theorem 2.1.1. Hypothesis 14.1.5 also includes the assump-
tion that there is a unique maximal 2-local Mc containing the centralizer in G of
Z := D1(Z(T)); that is,
Mc= !M(Cc(Z)).
The case where this condition fails will be treated in the final section of the chapter;
in that case Hypothesis 15.4.1.2 of the final section is satisfied.
By 14.1.12.1, there is M := M1 E M(T) - {Mc}, which is maximal under the
partial order:::_, on M(T) of Definition A.5.2, and Mis the unique maximal member
of M(T) - {Mc} under ::::.,_ As in Definition A.5.8, set V(M) := (ZM); as usual
V(M) E R 2 (M) by B.2.14.
The uniqueness theorems in A.5.7, for overgroups of T in M which coverM/CM(V(M)), replace the uniqueness theorems for members of £j(G, T), used
in the treatement of the Fundamental Setup (3.2.1), which are no longer available
as £1(G, T) is empty.
LEMMA 15.1.1. Set V := V(M), R := Cr(V), and M := AutM(V). Then
(1) Case (II) of Hypothesis 3.1.5 is satisfied with NM(R) in the role of "Mo",
and any H E H* (T, M).
(2) ij(M, V) ::; 2, and if q(M, V) > 2 then ij(Mi V) < 2.
1083