71Z 8. ELIMINATING SHADOWS AND CHARACTERIZING THE J4 EXAMPLEwhich are not SQTK-groups; as a consequence we obtain an improved bound on
r, and this slight improvement makes the remaining weak closure analysis much
easier. Since this improved bound on r now exceeds the value occurring in the
shadows, our calculations will in effect eliminate the Fischer groups-and in the
case of Mz 4 , will produce the centralizer of a 2-central involution resembling that
in J4.
In brief, we will use methods of pushing up to show for certain x E V that
Ca(x) :::; M. Consequently any U :::; V with Ca(U) i M must contain only
elements in conjugacy classes other than that of x. This restriction, added to those
from Table 7.2.1, produces the improved bound on r. Then the remaining weak
closure analysis proceeds rapidly.
In this section, we will by convention order the cases so that the case L ~ M 22
is first, the case L ~ Mz3 is second, and the case L ~ Mz4 is third. When we make
an argument simultaneously for all cases, we will list values of parameters for the
cases in that order, without explicitly writing "respectively". Thus for example,
the module Vis the cocode module, which we are denoting by 10, 11, 11.
We take the standard point of view (cf. section H.13 of Volume I) that the
cocode modules are sections of the space spanned by the 24 letters permuted by
Mz4, modulo the 12-dimensional subspace given by the Golay code. For Mz 4 , the
11-dimensional cocode module Vis the image of the subspace of all subsets of even
size. The orbits of Mz4 on V consist of the set Oz of images of 2-sets and the set
04 of images of 4-sets, with the latter determined only modulo the code-that is,
04 is in 1-1 correspondence with the sextets in the terminology of Conway [Con71]
and Todd [Tod66]. For Mz3 and Mzz we can consider 2-sets containi~g just one
of the letters fixed by this subgroup, and denote the corresponding vector orbit by
Oz.
Our subgroup M corresponds to a local subgroup M in the shadow group
G := Fzz, Fz3, Fz4. Notice in these shadows that for x E Oz, C 0 (x) i M; in
fact Ca(x) has a component, which is not strongly quasithin. We will see that the
results on pushing up in section 0.2 apply, and in fact rule out these components
which arise in the shadows, forcing Ca(x) :::; M.
PROPOSITION 8.1.1. Ca(x) :::; M for x E Oz.
PROOF. By H.15.1.1,C1(x) ~ Mz1, Mzz, Aut(Mzz),
where Mz1 ~ L3(4). Let H := Ca(x), MH := H n M, and LH := CL(x)^00 •
Replacing x by a suitable M-conjugate if necessary, we may assume TH:= Cr(x) E
Sylz(CM(x)). As F*(C1(x)) is simple, Oz(CL(x)TH) = Q = Oz(LT).
Next we show that Hypothesis C.2.8 is satisfied with Q, LH in the roles of "R,
Mo". Recall 'first that as part of the general setup in the introduction to chapter
7, C(G, Q):::; M. By A.4.2.7, Q is Sylow inCMH(LH/Oz(LH)), so that the second
hypothesis of C.2.8 is satisfied. By H.15.1.2, we have V = [V, LH] for the cocode