488 INTRODUCTION TO VOLUME II
which will emerge below) to work with a subgroup H which is minimal subject to
T :::; H, H i. M, and 02 (H) -=} 1. For example if G is a group of Lie type and
characteristic 2, then Mis a maximal parabolic over T, and H = 0
21
(P), where P
is the unique parabolic of Lie rank 1 over T not contained in M. Similar remarks
hold for other simple groups G with diagram geometries.
We introduce some further definitions to formalize this approach in our abstract
setting. We will need to work not only with 2-local subgroups, but also with various
subgroups of 2-locals, so we define
1i =Ha:= {H:::; G: 02(H)-=} 1},
and for X s;;; G, define 1-l(X) = 1-la(X) := {H E 1{ : X s;;; H}. Note that any
H E 1{ lies in the 2-local Na(0 2 (H)), and hence is contained in some member of
M. Thus as G is quasithin, each H E 1{ is in fact strongly quasithin; that is H
satisfies:
(SQT) mp(H) :::; 2 for each odd prime p.In addition each H E 1{ must also be a JC-group by our hypothesis (K), so H in
fact satisfies
(SQTK) His a JC-group satisfying (SQT).
The possible simple composition factors for SQTK-groups are determined in The-
orem C (A.2.3) in Volume I. The proof of the Main Theorem depends on generalproperties of JC-groups, but also on numerous special properties of the groups in
Theorem C, so we refer to the list of groups in that Theorem frequently through-
out our proof. We must also occasionally deal with proper subgroups which arenot contained in 2-locals. Such groups are quasithin JC-groups but not necessarily
SQTK-groups; thus we also require Theorem B (A.2.2), which determines all simple
composition factors of such groups.In view of Theorem 2.1.l, the set
1-l(T, M) :={HE 1-l(T) : Hi M}
is nonempty. Write 1-l*(T, M) for the minimal members of 1-l(T, M), partially
ordered by inclusion. Note that for HE 1-l*(T, M), H n Mis the unique maximal
subgroup of H containing T by the minimality of H. Further if Na(T) :::; M
(and we will show in Theorem 3.3.1 that this is usually the case), then T is not
normal in H. These conditions give the definition of an abstract minimal parabolic,
originating in work of McBride; see our definition B.6.1. The condition stronglyrestricts the structure of H. In particular, the possibilities for H are described
in sections B.6 and E.2. In the most interesting case, 02 (H/0 2 (H)) is a Bender
group, so H does resemble a minimal parabolic in the Lie theoretic sense for a group
of Lie type: namely 021 (P) where Pis a parabolic of Lie rank 1.
Thus for each ME M(T), we can choose some HE 1-l*(T,M). By the maxi-mality of M, (M, H) is not contained in a 2-local subgroup, so that 02 ( (M, H)) = 1.
Thompson's weak closure methods and the later amalgam method depend on the
latter condition, rather than on the maximality of M, so often we will be able
to replace M by a smaller subgroup. We say U is a uniqueness subgroup of G if
:J!M(U). Furthermore we usually write M = !M(U) to indicate that M is the
unique overgroup of U in M. Notice that if M = !M(U), then from the defini-