Part 1 has set the stage for the proof of the Main Theorem by supplying infor-
mation about the structure of 2-locals, establishing the Fundamental Setup (3.2.1),
and proving that in the FSU, the members of H*(T, M) are minimal parabol-
ics. We now begin the analysis of the various possibilites for L E .Cj(G, T) and
VE R 2 (L 0 T) arising in the FSU. Recall the FSU includes the hypotheses that G is
a simple QTKE-group, TE Syl 2 (G), and LE .Cj(G, T) with L/0 2 (L) quasisimple
and V a suitable member of R 2 (LT).
In Part 2, we consider the Generic Case of our Main Theorem. This is the
case where L/0 2 (L) ~ L 2 (2n) with L ~Mand n(H) > 1 for some HE H*(T, M).
We show in Theorem 5.2.3 of chapter 5 that in the Generic Case, (modulo the
sporadic exception M 23 and the "F2-case") G is one of the generic conclusions in
our Main Theorem: namely G is of Lie type of Lie rank 2 and characteristic 2. In
chapter 6 we consider the remaining case where n(H) = 1 for each HE H*(T, M),
and show in that case that n = 2 and Vis the A 5 -module. The case where V is
12.2. Groups over F 2 , and the case V a TI-set in G
module for n4(2).
Thus once we have dealt with the groups L 2 (p) and the Bender groups in
Theorem 2.1.1, and the groups of Lie type in characteristic 2 of Lie rank 2 in
Theorem 5.2.3, we will have handled all the infinite families of groups appearing as
conclusions in the Main Theorem.