0.3. AN OUTLINE OF THE PROOF OF THE MAIN THEOREM 493In order to discuss these cases in more detail, we need more concepts and
notation.
First, another consequence of Theorem 3.1.1 (established as part (3) of Theorem3.1.8) is that either
(i) L = [L, J(T)], or
(ii) H*(T, M) ~ Cc(Z), where Z = fh(Z(T)).
Here J(T) is the Thompson subgroup ofT (cf. section B.2). In case (i), Vis an FF-
module; so when Vis not an FF-module, we know [Z, H] = 1 for all HE H*(T, M).
In particular Cv(L) = 1 since His not contained in the uniqueness group M for LT,whereas if Cv(L) were nontrivial then Cz(L) would be nontrivial and centralized
by H as well as LT.
Second, in section E.l, we introduce a parameter n(H) for H E H. The pa-
rameter involves the generation of H by minimal parabolics, but the definition of
n(H) is somewhat more complicated; for expository purposes one can oversimplify
somewhat to say that roughly n(H) = 1 unless H has a composition factor which
is of Lie type over F 2 n-in which case n(H) is the maximum of such n. Thus for
example in a twisted group H of Lie type, n(H) is usually the exponent n of the
larger of the orders of the fields of definitions of the Levi factors of the parabolics
of Lie rank 1 of H. In particular if HE H*(T, M), then either n(H) = 1, or (using
section B.6) 02 (H/0 2 (H)) is a group of Lie type over F 2 n of Lie rank at most 2,
02 (H) n M is a Borel subgroup of 02 (H), and n(H) = n. In that event, we call
the Hall 21 -subgroups of H n M Cartan subgroups of H. Our object is to show
that n(H) is roughly bounded above by n(L), and to play off Cartan subgroups
of H against those of L when L/0 2 (L) is of Lie type. It is easy to see that if
n(H) > 1 and Bis a Cartan subgroup of H n M, then H = (H n M, NH(B)), so
that Nc(B) 1:. M. On the other hand, if n(H) is small relative to n(L) (e.g. if
n(H) = 1), then weak closure arguments can be effective.
Third, except in certain cases where V is a small FF-module, we obtain the
following important result, which produces still more uniqueness subgroups:.
Theorem 4.2.13 With small exceptions, if I:::; LT with L :::; 02 (LT)I and
I E H, then I is also a uniqueness subgroup.Theorem 4.2.13 has a variety of consequences, but perhaps its most important
application is in Theorem 4.4.3, to show that (except when Vis a small FF-module)
if 1 =/= B is of odd order in CM(V), then Nc(B) :::; M. In particular from the
previous paragraph, if HE H*(T, M) with n(H) > 1 and Bis a Cartan subgroup
of H n M, then [V, BJ =!= 1. If [Z, H] = 1, this forces B to be faithful on L, so that
it is possible to compare n(H) to n(L) and show that n(H) is not large relative to
n(L).
0.3.4.l. Weak Closure methods. Thompson introduced weak closure methods
in the N-group paper [Tho68]. When n(H) is small relative to n(L) and (roughly
speaking) q(LT/0 2 (LT), V) is not too small, weak closure arguments become ef-