CHAPTER 5
The Generic Case: L 2 (2n) in £1 and n(H) > 1
In this chapter we assume the following hypothesis:
HYPOTHESIS 5.0.1. G is a simple QTKE-group, TE Syl 2 (G), L E Cj(G, T)with L/02(L) ~ L2(2n) and L :SIME M(T).
As Lis nonsolvable, n ~ 2. Further M = !M(LT) by 1.2.7.3 and M = Na(L).
SetZ := Di(Z(T)).
From the results of section 1.2, there exists V E R 2 (LT) with [V, L] -=/=-1; choose
such a V and set LT:= LT/CLT(V). By 3.2.3 it is possible to choose V so that
the pair L, V satisfies the hypotheses of the Fundamental Setup (3.2.1). However
occasionally we need information about other members of R 2 (LT), so usually in
this chapter we do not assume V satisfies the hypotheses of the FSU. Later, whenappropriate, we sometimes specialize to that case.
By Theorem 2.1.1, 'H*(T, M) is nonempty.
In the initial section 5.1, we determine the possibilities for V and provide
restrictions on members of 'H* (T, M). The following section begins the proof of
Theorem 5.2.3, which supplies very strong information when n(H) > 1 for some
HE 'H*(T, M). Indeed in the FSU, if Vis not the A 5 -module, then either G is of
Lie type and Lie rank 2 over a field of characteristic 2, or G is M 23 ; hence we refer
to this situation as the Generic Case. The final section 5.3 completes the proof of
Theorem 5.2.3.
Our primary tool for proving Theorem 5.2.3 is the main theorem of the "Green
Book" of Delgado-Goldschmidt-Stellmacher [DGS85], which gives a local descrip-
tion of weak EN-pairs of rank 2. To apply the Green Book, we must achieve the
setup of Hypothesis F.1.1. There are two major obstacles to verifying this hypoth-
esis: Let D be a Hall 2'-subgroup of NL(T n L), and K := 02 (H). We must first
show that D acts on K, unless the exceptional case in part (1) of Theorem 5.2.3
holds. Second, we must construct a normal subgroup S of T such that S is Sylow
in SL and SK, and so that there exists an S-invariant subgroup Ki of K such that
K 1 /0 2 (K 1 ) a Bender group. Now K/0 2 (K) is of Lie type in characteristic 2 of Lie
rank 1 or 2. If K is of Lie rank 1, we take Ki := K; if K is of Lie rank 2, we choose
Ki to be a rank one parabolic of K. In either case, we take S to be 02(H n M),
unless K/0 2 (K) ~ L 3 (4), which provides a final obstruction that we deal with in
Theorem 5.1.14.
After producing our weak EN-pair and identifying it up to isomorphism of
amalgams using the Green Book, we still need to identify G. To do so we appeal
to Theorem F.4.31 as a recognition theorem; ultimately Theorem F.4.31 depends
upon the Tits-Weiss classification of Moufang generalized polygons, although the
Fong-Seitz classification of split EN-pairs of rank 2 would also suffice. There is also
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