10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 7Slthe other hand if I~ Sp4(2n), then Lis indecomposable on 02 (L), so V 1 = 02 (L).
Then there is X S N1 ( L) of order 2n -1 centralizing L /V 1 and faithful on Vi. Thus
XS Na(L) SM, so XS Lo by 10.1.3, impossible as there is no such subgroup of
Lo.Thus I is not quasisimple. So E(I) = 1 by A.3.3.1. We claim F*(ITv) =
02(ITv): If not, then O(I) i=-1 as E(I) = 1. But by A.1.26.1, V 1 = [V 1 , L] central-
izes O(I), so O(I) SM by 10.2.6.2, and hence O(CM(v)) i=-1, a contradiction asCM(v) E 1-{e by 1.1.3.2.
We have shown that F*(ITv) = 02(ITv)· So VJ := (CVi (Tv)^1 ) E R2(ITv)
by B.2.14. Let (ITv)* := ITv/CIT,,(VJ). Now Vv := (CVi(Tv)L) S V1, and from
the action of Lo on V in 10.1.1, either Vi = V,, or case (3) of 10.1.1 holds with
Cv 1 (Tv) = Cv 1 (Lo) i=- 1 and V,, = Cv 1 (Lo). Therefore either Cv(Lo) i=- 1, orNa(Vv) S M by 10.2.6.2. In the former case, 1 i=-Cz(Lo) S VJ, so Ca(VJ) S
Ca(Cz(Lo)) SM= !M(LoT); in the latter, Ca(VJ) S Ca(Vv) S M. So in any
case, Ca(VJ) SM, and hence L* <I* as Ii. M, while L* i=-1 as I= (L^1 ).
Next observe that J(T) S Tv, so that J(T) = J(Tv) and S = Baum(Tv): IfJ(T) S CT(V) this is clear, so by 10.1.2.1 we may assume that one of the first three
cases of 10.1.1 holds. But in each of these cases v centralizes some M-conjugate ofJ(T), so again the remark holds.
We next claim that I = [I, J(Tv)*] is quasisimple. Suppose not, so that either
[VJ; J(Tv)] = 1 or I is not quasisimple. Suppose first that J(Tv) i=-1. Then
I* is not quasisimple, so I* is described in case ( c) or ( d) of 1.2.1.4, and hence
[X*, J(Tv)*] i=-1 for X := Sp(I) and some prime p > 3, contradicting SolvableThompson Factorization B.2.16. Thus we may take J(Tv) = 1. However L i=-1,
so J(T) S 02 (LTv) and hence J(T) ::::! LoT, so that Na(J(T)) SM. Then by a
Frattini Argument, I= C1(VJ)N1(J(T)) SM, contradicting Ii. M. So the claim
is established.By the claim, VJ is an FF-module for I*T;. Now intersecting the list of pos-
sibilities for the embedding of L in I in A.3.12 with the list of B.4.2, we get the
following cases:
(a) L ~ L2(2n), I*~ SL3(2n), Sp4(2n), or G2(2n), and 02(L*)-=/:-1.
(b) L ~As ·or L3(2), and I*~ A1 with 02(L*) = 1.
(c) L ~ L 3 (2) and I*~ L4(2) or Ls(2), with 02(L*) -=f-1.
Observe in particular that I does not appear in case (c) or (d) of 1.2.1.4, so J/02(I)
is quasisimple.Assume case (a) holds. Recall we saw earlier that Vi = Vv S VJ and the
FF-module VJ is described in Theorem B.5.1. Then L = N1(Vi)^00 and N1* (Vi)
is a maximal parabolic of I*, so N1(L) contains a subgroup X of order 2n - 1
centralizing L / 02 ( L) and nontrivial on Vi. We now get a contradiction much as
in the earlier case of Sp4(2n) where I was quasisimple: for XS Na(L) SM, and
hence XS Lo by 10.1.3, whereas there is no such subgroup of Lo.
Thus we have shown that (b) or (c) holds, so L ~As or L3(2). We next show:
In case (b) either