CHAPTER 10
The case L E £j ( G, T) not normal in M.
In this chapter we prove:THEOREM 10.0.1. Assume G is a simple QTKE-group, T E Syh(G), and
LE .Cj(G, T) with L/02(L) quasisimple. Then T:::; Na(L).
10.1. PreliminariesAssume Theorem 10.0.1 is false, and pick a counterexample L. Let Lo :=(LT)
and M := Na(Lo). By 3.2.3, there is Va E Irr +(L 0 , R 2 (L 0 T)) such that L and
Vr := (Vt) are in the Fundamental Setup 3.2.1. Set V-:= (V,P), and note that thisdiffers from the notation in the FSU where Vr, V are denoted by "V, VM". Note in
particular that by construction V ::;I M, so that M = NG (V).
As L <Lo, we can appeal to Theorem 3.2.6. In the first two cases of Theorem
3.2.6, Vr is not an FF-module, and those cases were eliminated in Theorem 7.0.1.Thus we are left with case (3) of Theorem 3.2.6. We recall from that result that
V =Vi V{ fort ET-Nr(L), with V 1 := [V, L]:::; Cv(Lt).
Recall that in the FSU with V ::;I M, we set M := M/CM(V) and V =
V/Cv(Lo). Also set Li:= L, Lz :=Lt fort ET-Nr(L), and Vi:= [V, Li]·
The cases to be treated are listed in the following lemma. Subcases (ii) and(iii) of 3.2.6.3 appear as cases (5) and (6) in 10.1.1. In subcase (i) V 1 E Irr +(L, V),
and by 3.2.6.3b, q(AutL 0 r(V1), V1) :::; 2, so V appears in B.4.2 or B.4.5. As L <Lo,
L appears in 1.2.1.3. Intersecting those lists leads to the remaining cases in 10.1.1.
LEMMA 10.1.1. V = Vi Vz E R2(M) with Vi := [V,Li] :::; Cv(L3-i), V =
V1 E8 V2, and one of the following holds:
( 1) V1 is the natural module for L ~ L 2 ( 2n), with n > 1.
(2) V1 is the A5-module for L ~ A5·
(3) V 1 is the natural module for L ~ L 3 (2).(4) Vi is the orthogonal module for L ~ 04(2n), with n > 1.
(5) V1 is the sum of a natural module for L ~ L3(2) and its dual, with the
summands interchanged by an element of Nr(L).
(6) V1 is the sum of four isomorphic natural modules for L ~ L 3 (2), and
02 (CM(L)) ~ Z5 or E25.
(7) V 1 is the natural module for L ~ Sz(2n).
Let Z := 01(Z(T)), to := T n Lo, T1 := Nr(L); and Bo := 02 (NL 0 (To)).
Note that B 0 T = TB 0 and (except when L ~ L 3 (2) where Bo= 1) Bo is a Borel
subgroup of L 0. Set S := Baum(T).
LEMMA 10.1.2. (1) Except possibly in the first three cases of 10.1.1, V is not
an FF-module for AutL 0 r(V), so J(T) :::; Cr(V). ·
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