S.2. USING WEAK BN-PAIRS AND THE GREEN BOOK 649we obtain p = 3; notice that here L2/02(L 2 ) is not J 1 , since here p = 3 or 5 rather
than 7. Since B acts on D3 by 5.1.5.2, B centralizes D 3. But also B ::::; Na(L 2 ) so
D3 1:. B; hence M;::: D3B ~ E27, a contradiction as Mis an SQTK-group. This
completes the proof of 5.2.2. DWe now state the main result of this chapter:
THEOREM 5.2.3. Assume G is a simple QTKE-group, TE Syh(G), and LE
.Cj(G,T) with L/02(L) ~ L 2 (2n) and L ::::1 M E M(T). In addition assume
HE H*(T,M) with n(H) > 1, let K := 02 (H), Z := fh(Z(T), and VE R 2 (LT)
with [V, L] #-l. Then one of the following holds:
(1) n = 2, Vis the sum of at most two copies of the As-module for L/0 2 (L) ~
As, and K::::; Kz E C(Ca(Z)). Further either K/02(K) ~ L 2 (4) with Kz/0 2 (Kz)
~ A1, J2, or M23, or K = Kz and K/02(K) ~ L3(4).
(2) G ~ M23·(3) G is a group of Lie type of characteristic 2 and Lie rank 2, and if G is
Us(q) then q = 4.Note that conclusions (2) and (3) of Theorem 5.2.3 are also conclusions in our
Main Theorem. Thus once Theorem 5.2.3 is proved, whenever LE .Cj(G, T) is T-
invariant with L/0 2 (L) ~ L 2 (2n), we will be able to assume that either conclusion
(1) of Theorem 5.2.3 holds, or n(H) = 1 for each HE H*(T, Nc(L)). The treatment
of these two remaining cases is begun in the following chapter 6, and eventually
completed in Part 5, devoted to those LE .Cj(G, T) with L/0 2 (L) defined over F 2.5.2.1. Determining the possible amalgams. The proof of Theorem 5.2.3
will not be completed until the final section 5.3 of this chapter. In this subsection,
we will produce a weak BN-pair a, and use the Green Book [DGS85] to identify
a up to isomorphism of amalgams. This leaves two problems: First, show that the
subgroup Go generated by the parabolics of a is indeed a group of Lie type. Second,
show that Go = G. In qne exceptional case, Go is proper in G; the second subsectionwill give a complete treatment of that branch of the argument, culminating in the
identification of G as M23·Assume the hypotheses of Theorem 5.2.3. Notice that Hypothesis 5.1.8 holds,
since in Theorem 5.2.3 we assume n(H) > l. During the proof of Theorem 5.2.3,
write D for DL.
Notice that if K/0 2 (K) ~ L 3 (4), then conclusion (1) of Theorem 5.2.3 holds
by Theorem 5.1.14. Thus we may assume during the remainder of the proof of
Theorem 5.2.3 that K/0 2 (K) is not £ 3 (4). Therefore by 5.1.11.3, S acts on the
rank one parabolics of K, and hence on the group L2 defined at the start of the
section.
Next if D 1:. Nc(K), then conclusion (2) of 5.2.2 is satisfied, so again conclusion
(1) of Theorem 5.2.3 holds. Thus we may also assume during the remainder of the
proof that D acts on K; we will show under this assumption that conclusion (2) or
(3) of Theorem 5.2.3 holds. The following consequences of these observations are
important in producing our weak BN-pair:
LEMMA 5.2.4. (1) D::::; Na(K).
(2) D::::; Nc(B) and B::::; Na(D).
(3) B::::; Nc(S), D::::; Nc(S n L2), and DS =SD.
(4) DSB acts on L2·