490 INTRODUCTION TO VOLUME TI
with P the middle node maximal parabolic over T n Go, and H := PT. Then
H ~ (L,T) for an LE C(G,T) with ILTI = 2.
We partially order £(G, T) by inclusion and let C*(G, T) denote the maximal
members of this poset. In our earlier example where H is a parabolic of a group of
Lie type, notice that any LE C(H) is contained in a maximal parabolic determined
by some end node. Thus the C-components of such parabolics are the members of
C*(G, T).
In an abstract QTKE-group G, the members of £,*(G,T) can be used to pro-
duce uniqueness subgroups: For by 1.2.4, when S E Syb(H), any L E C(H, S) is
contained in some KE C(H). Then a short argument in 1.2.7 shows that whenever
LE C*(G,T),
Na((LT)) =!M((L,T)).
Thus (L, T) is a uniqueness subgroup in our language, achieving the goal of this
subsection.
But it could also happen (for example in a group of Lie type over the field
F 2 ) that the visible 2-locals are solvable, so that £( G, T) is empty. To deal with
such situations, and with the case where L/0 2 (£) is not quasi.simple for some
LE £,*(G,T), we also show that certain solvable minimal T-invariant subgroups
are uniqueness subgroups. The quasi.thin hypothesis allows us to focus on p-groups
of small rank: Define 3(G, T) to consist of those T-invariant subgroups X = 02 (X)
of G such that ·
XT E Ji, X/0 2 (X) ~ Ep2 or p1+^2 for some odd prime p, and Tis irreducible
on the Frattini quotient of X/02(X).
For example, in the extension of L4(2n) discussed above, if we taken= 1 instead
of n > 1, .then H =PT E 3(G, T) for p = 3.
If X is not contained in certain nonsolvable subgroups, then XT will be a
uniqueness subgroup. Thus we are led to define 3*(G, T) to consist of those X E
3(G, T) such that XT is not contained in (L, T) for any LE £(G, T) with L/0 2 (£)
quasi.simple. We find in 1.3. 7 that if X E 3* ( G, T), then
Na(X) =!M(XT),
so that XT is a uniqueness subgroup.
0.3.3. Classifying the uniqueness groups and modules. We now return
to our pair M, H with M E M(T) and HE 1i*(T, M) from subsection 0.3.1. The
structure of H is restricted since H is a minimal parabolic, but a priori M could
be a fairly arbitrary quasi.thin group, subject to the constraint F*(M) = 02 (M);
in particular, the composition factors of M could include arbitrary simple SQTK-
groups acting on arbitrary "internal modules" (elementary abelian .M-sections)
involved in 02 ( M)
To obtain a more tractable set of possibilities, we exploit a uniqueness subgroupU produced by one of the two methods in the previous subsection 0.3.2; that is, we
take U of the form (L,T) with LE C(G, T), or XT with XE 3(G, T), and take
M := Na(0^2 (U)) = !M(U). Recall that Z := !1 1 (Z(T)) cannot be central in both
U and H. The case where Z :::; Z(U) for all choices of U is essentially a "small"
case, treated in Part 6, so most of the analysis deals with the case [ Z, U] # 1.