714 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLE
If m(.A) = 5, then by H.14.3.1, A= KQ· Then W = V by H.15.4.4, contrary to
the previous paragraph. Thus m(.A) = 4, so as W < V, H.15.5 says m(V/W) ~ 1.
But by earlier remarks, w = 2 and m(V/W) ~ w. This contradiction completes
the proof of the Theorem. D
8.2. Determining local subgroups, and identifying J4
In this section we treat the remaining cases other than L3(2) wr Z2, which
we consider in the final section of the chapter; thus in addition to Hypotheses 7.0.2
and 7.3.1, we assume:
HYPOTHESIS 8.2.1. L 0 is not L 3 (2) x L3(2) on the tensor module 9.
As a result of the previous section, we have eliminated M22 and M23 on their
cocode modules, and in the case of M24 on its cocode module, we showed there is
a unique solution for the weak closure parameters of aw-offender A on V. Indeed
in that case we showed that A= KT and Cv(A) = Cv(KT) is of dimension 3.
Because of Hypothesis 8.2.1, the other cases to be treated in this section are:
L ~ (S)L3(2^2 n)/9n, M22/l0, M24/ll.
As before we will use this ordering in common arguments, and we adjoin M24/ll
as the fourth case on our list. In the first three cases we will show (as we did
in case four) that there is a canonical choice for our w-offender A, and for each
such canonical A, Cv(A) is determined. Then in all four cases, we construct a
sizable part of the local subgroup N := Na(Cv(A)). In some cases N will not be
strongly quasithin, so those cases are eliminated. In the surviving cases we study
C := Co(z), where z is a 2-central involution in V; from CM(z) and CN(z) we
can construct enough of C to see that either C is not strongly quasithin, or that
M ~ M 24 /ll and C has the structure of the centralizer of an involution in J4.
Then we identify G as J4 in the final subsection of this section.
8.2.1. Isolating a w-offender. As usual let H E H*(T, M). Recall His a
minimal parabolic by 3.3.2.4, with H n M the unique maximal overgroup of Tin
H. We see in the next lemma that V f:_ 02(H), so from lemma E.2.9, the set
I(H, T, V) of Definition E.2.4 is nonempty.
PROPOSITION 8.2.2. (1) V "i. 02(H).
(2) There exists h E H such that I := (V, Vh) is in the set I(H, T, V) and
h E J.
(3) 1 :f: Zr:= v n vh s Z(I).
(4) Tr:= T n IE Syl2(I) and Mr:= Mn I= Nr(V).
(5) kerMAI) = 02(I) 1 and I*:= I/02(I) ~ D2mi m odd (in which case we setk := 1), L2(2k), or Sz(2k), for some suitable k dividing n(H).
(6) V* = Z(Tj) and Mj = Nr.(Tj).
(7) A:= Vhn02(I) = Nvh(V), CA(V) =Zr, A is cubic on V, rAutA(V),V < 2,
m(A) = m(V/Zr) - k, and Cv(A) s B := V n 02(I).
(8) If k > 1, then Cv(X) =Zr for X of order 2k -1 in Mr.
PROOF. From Table B.4.5, either Mv = Aut(M22) and Vis the code module;
or q(Mv, V) > 2, so that V f:_ 02(H) by 3.1.8.2, and (1) holds.
Therefore we may assume that V s 02 (H) with V the code module for