742 10. THE CASE LE .Cj(G, T) NOT NORMAL IN M.
. (2) J(T) ::; NT(L) =Ti.
(3) CT(V) = 02 (L 0 T) except' in case (6) of 10.1.1, where at least CT(V) <I
L 0 T. In any case, M = !M(Na(CT(V))) = !M(Na(J(CT(V)))).
(4) Except possibly in cases (1) and (3) of 10.1.1, Cv(Lo) = 1.
(5) Assume L is not L 3 (2) and let D be a Hall 2'-subgroup of Bo. Then either:
(a) CD(Z) = 1, or
(b) Vi is the orthogonal module for L 2:! n4(2n) and CD(Z) 2:! Z~n+i·(6) In cases (1) and (2) of 10.1.1, L 0 T is a minimal parabolic in the sense of
Definition B.6.1, with NLoT(To) the unique maximal overgroup of Tin LoT. Thus
if J(T) -f:. CT(V) then L 0 T is described in E.2.3 and S::; NT(L) =Ti.
PROOF. Part (2) is clear if J(T) ::; CT(V), while if J(T) i CT(V), it followsfrom B.1.5.4. Except in cases (5) and and (6) of 10.1.1, VJ. is an irreducible for L, and
(1) follows from B.4.2. In cases (5) and (6), Vis not an FF-module for AutL 0 T(V)
by Theorem B.5.6, so (1) is established. Next in all cases of 10.1.1 except case (6),V = VT, so that CT(V) = 02(LoT) by 1.4.1.4. In case (6), CL 0 T(V) ::::; 02(LoT),
so as V :::l L 0 T, CT(V) = CLoT(V) :::! L 0 T, and hence (3) holds in that case too.
Part (4) follows in the final four cases of 10.1.1 from (1) and 3.2.10.9; in the second
case it follows from I.1.6. Part (5) follows easily from (4) and the structure of the
modules in 10.1.1. Finally the first two remarks in (6) are elementary observations,and then if J(T) -f:. CT(V), the remaining remarks are a consequence of E.2.3. D
LEMMA 10.1.3. Lo= QP' (M) for each prime divisor p of ILi.
PROOF. This follows from 1.2.2. D10.2. Weak closure parameters and control of centralizers
We will make use of weak closure, together with .control of centralizers of el-
ements of vt. In 10.2.3, we will use the fact that G is a QTKE-group to show
n(H) ::; 2 for HE 1-i*(T, M); subsequent results provide lower bounds on the weak-closure parameters r(G, V) and w(G, V). In 10.2.13, we will eliminate most cases
using the relation n(H) 2': w(G, V) in E.3.39. ·
LEMMA 10.2.1. Except possibly in case (3) of 10.1.1, Na(S) ::; M. ·
PROOF. We may assume case (3) of 10.1.1 does not hold. If J(T) ::; CT(V),
then as J(T)::::; S, Na(S)::::; M by 3.2.10.8. Thus we may assume J(T) -f:. CT(V),
so by 10.1.2.1, we are reduced to cases (1) and (2) of 10.1.1. In those cases, L 0 T is
a minimal parabolic, and is described in E.2.3 by 10.1.2.6..
In case (1) of 10.1.1, E.2.3.2 says S E Syb(L 0 S), so we can apply Theo-rem 3.1.1 with LoT, Na(S), S in the roles of "H, Mo, R", to conclude that
02( (Na(S), LoT)) # 1. Thus Na(S) ::; M = !M(LoT), as desired.
Therefore we may assume case (2) of 10.1.1 holds; the proof for this case will
be longer. Moreover for each S+ :::) T with To = T n L ::; S+, S+ E· Syl2 (LoS+);hence applying 3.1.l as in the previous paragraph, we conclude that Na(S+) ::;
M. In particular Na(Ti) ::; M. We may also assume that Na(S) -f:. M, so as
M = !M(LoT), no nontrivial characteristic subgroup of Sis normal in LoT. Then
E.2.3.3 says that Li is an As-block.
Suppose first that Cz(Lo) = 1. Then 02 (LoT) = V by C.l.13.c, so that
V = 02(M) using A.1.6. Further using E.2.3, S =Six 82, where Si := Cs(L3-i) =