634 5. THE GENERIC CASE: L2(2n) IN .Ct AND n(H) > 1permutes {DiZ(8), D 2 Z(8)}. Then 02 (Na(8)) acts on DiZ(8), and indeed cen-
tralizes DiZ(8)/Z(8) as DiZ(8)/Z(8) is of order 4 and contains a unique coset
of Z(8) containing elements of order 4. Thus 02 (Na(8)) acts on Q, and hence
02 (Na(8)) SM= !M(Na(Q)). But then Na(8) = 02 (Na(8))T SM, contrary
to assumption. This contradiction completes the proof of (1).
As (1) is established, we may assume the hypotheses of (2). Thus [Z,H] '/:-1,
so J(T) i CT(V) by 3.1.8.3, and then part (i) of (2) holds by B.6.8.6.d. Therefore
by 5.1.2, either [V, L] is the 85 -module for LT ~ 85, or V/Cv(L) is the natural
module for L. Set U := (ZR), so that U E R2(H) by B.2.14. By (1), 8 #
Baum(02(H)). Then as [Z, H] '/:-1, J(T) i CT(U) by B.6.8.3.d, and (ii) follows.Finally if 02( (NL(T n L), H)) = 1, we may apply 5.1.6; as [Z, HJ '/:-1, conclusion
(3) of 5.1.6 holds, completing the proof of (iii). D5.1.2. Further analysis when n(H) > 1. Recall that in this Part we focus
on the "generic" situation, where n(H) > 1 for some H E 1i*(T, M). Later in
Theorem 6.2.20, we will reduce the case where n(H) = 1 for each HE 1i*(T, M) to
n = 2 with L = L 2 (4) ~ A5 acting on [Z, L] as the sum of at most two A 5 -modules.
That situation is treated later in those Parts dedicated to groups defined over F 2.
So in the remainder of this section we assume the following hypothesis:HYPOTHESIS 5.1.8. Hypothesis 5.0.1 holds, and there is H E 1i*(T, M) with
n(H) > 1. Set K := 02 (H), MH :=Mn H, and MK :=Mn K.
NOTATION 5.1.9. By 5.1.5.2, we may choose a Hall 2'-subgroup B of MH, anda B-invariant Hall 2'-subgroup DL of NL(TnL). This notation is fixed throughout
the remainder of the section.
Observe MH =BT= TB since TE 8yh(MH)· Further Band T normalize
NL(T n L) = DL(T n L) by 5.1.5.1, so DLBT is a subgroup of G.
Our goal (oversimplifying somewhat) is to show in the following section that
(LTB,DLTB,DLH) forms a weak EN-pair of rank 2 in the sense of [DGS85], as
in our Definition F.1.7. Indeed we already encounter such rank 2 amalgams in this
section.
The next few results study the structure of K and the embedding of K in mem-
bers X of 1i(H), and show that usually DL n X acts on K. This last type of result
is important, since to achieve Hypothesis F.1.1 and show (LTB, TDLB, HDL) is a
weak EN-pair of rank 2, we need to show DL acts on K.
LEMMA 5.1.10. Let k := n(H) and H := H/0 2 (H). Then K is a group of
Lie type over F 2 k of Lie rank 1 or 2, MK is a Borel subgroup of K, and B is a
Cartan subgroup of K*. More specifically, K = (K'[) for some Ki E C(H), and
one of the following holds:
(1) Ki < K and Ki~ L2(2k) or Sz(2k).
(2) Ki = K and K* is a Bender group over F 2 k.
(3) Ki = K, K* ~ (8)L 3 (2k) or 8p 4 (2k), and T is nontrivial on the Dynkin
diagram of K*.PROOF. As n(H) > 1, this follows from E.2.2. D
From now on, whenever we assume Hypothesis 5.1.8, we also take K 1 E C(H).
LEMMA 5.1.11. Let 8 := 02(MH) and H* := Hj0 2 (H). Then