578 3. DETERMINING THE CASES FOR LE .C.j(G, T)
A.l.31.1, s :::; 2; by E.2.3.2, if Y/ is a symmetric group, then Y/ is 83 or 85; and
in any case H n M is the product of T with the preimages of the Borel subgroups
over T n 1j in 1j*. Further ifs= 2, then as Ui is irreducible under H, {Ui,1, Ui,2}
is permuted transitively by T.
Assume for the moment that 1j ~ L 2 (2n) with n > 1 and some Ui,j the natural
module. Then by B.4.2.1, A is Sylow either in Y or in some Yj. Now A* is also
an FF -offender on U 3 _i, and B.4.2 says that the only other possible FF -module
for Yj is the A 5 -module when n = 2, whereas the FF -offenders on that module
are not Sylow in 1j*. Thus in any case U 1 is Y-isomorphic to U2.
Let K := 02 (H), and Wan H-submodule of UH maximal subject to Uo :=
[UH,K] f:. W. Set Ujf := UH/W. Thus Uft =f. 0, His irreducible on Uft, and
Cu+(K) = 0. As UH= (VH), Ujf = (V+Hj, so V 0 + := Cv+(T) =f. 0. As Cu+(K) =
H H
0, v 0 + :::; Uft using Gaschiitz's Theorem A.l.39. As H is irreducible on Uft, we may
take U1 = Uft. Further
0 =f. v 0 + :::; Cu 1 (J(R)*),
and as case (II) of Hypothesis 3.1.5 holds,
H n M acts on v+. (**)
Let X denote a Cartan subgroup of }j n M.
Suppose that 1j* ~ L 2 (2n) with n > 1 and U 1 ,j the natural module. Then as
J(R) E Syb(Y), we conclude from (*) and (**) that
Vj+ := v+ n U1,; = Cu 1 ,j ( J(R)*) (!)
is the J(R)*-invariant 1-dimensional F 2 n-subspace of Ui,j· In particular X acts
faithfully on V. This is a contradiction to the hypotheses of (5), and under the
hypotheses of (6), 03 (X) = 1 son= 2. But now Vj+ is the only noncentral chief
factor for X on v+, and the image of [V n W, X] in U 2 ,j is contained in Cu 2 ,j ( J ( R) *),
so X has a single noncentral chief factor on V n W. Thus X has just two noncentral
chief factors on V, contrary to the hypotheses of (6).
We have completed the proof of (5), so we may assume the hypotheses of (6)with Ui,j the natural module for Yj* ~ 83 or 85. By (4) and the hypothesis of (6),
m(A) ~ m ~ 2, so H is not 83. Thus we may assume 1j* ~ 85. Then from
the description of FF-offenders in B.3.2.4, 02 ((HnM)) = [0^2 (HnM), J(R)],
so as H n M acts on V and J(R) centralizes V,.X centralizes V, contrary to the
hypotheses of (6). This completes the proof of (6). D
3.2. The Fundamental Setup, and the case division for Lj(G, T)
The bulk of the proof of the Main Theorem involves the analysis of variouspossibilities for L E Lj(G,T). In this section we establish a formal setting for
treating these subgroups, and provide the list of groups L and internal modules V
which can arise in that setting. In the language of the Introduction to Volume II,
this gives a solution to the First Main Problem-reducing from an arbitrary choice
for L, V to the much shorter list arising in what we call below our Fundamental
Setup (FSU).
In this section we assume G is a simple QTKE-group, T E Syl 2 (G), Z :=
ni(Z(T)), and ME M(T). The notation Irr +(X, V) and Irr +(X, V, Y) appears
in Definition A.l.40. We will be primarily interested in