CHAPTER 8
Eliminating shadows and characterizing the J 4
exampleWe begin by reviewing the cases remaining after the work of the previous
chapter, which eliminated those cases which do not lead to examples or shadows.
We continue to assume Hypotheses 7.0.2 and 7.3.1 from the previous chapter.
The latter hypothesis excludes the case where Lo is nt (2n) on its orthogonal mod-ule; that case will be treated in chapter 9 of this part, because the methods used
to attack that case are different from those in the remaining cases.
The cases Lo/V remaining from Table 7.1.1 that were not eliminated in the
previous chapter 7, and are not among the cases to be treated in later chapters, are:L3(2^2 n)/9n, M22/l0 or M22/fO, M23/fl, M24/ll or M24/ll, and (L3(2) 12)/9.
In the case of (L3(2)12) /9, technical complications also arise, primarily becausethe existence of small (F-1)-offenders on V only gives r ::'.:'. 3. As a result, different
methods are required to treat this case; thus we will defer its treatment to 8.3.1 in
the final section of this chapter.As indicated in Table 7.1.1 in the previous chapter, the subgroups M we study
in this chapter do arise as maximal 2-locals in various shadows, and in the case of
M24 on its cocode module 11, in ·the quasithin example J4. Thus we should not
expect the methods of the previous chapter to eliminate these configurations onsimple numerical grounds. Instead we seek to show that our bounds determine a
unique solution for the various parameters: namely, the solution corresponding tothe shadow or example. Then to eliminate the shadows, we go on to show that this
unique solution leads (via study of w-offenders and subgroups H E 1i*(T, M)) to
a local subgroup other than M which is not an SQTK-group. In the M 2 4/ll case,
we construct the centralizer of a 2-central involution, which allows us to identify G
as J4.8.1. Eliminating shadows of the Fischer groups
In this section, we assume Lis M22, M23, or M 24 and Vis the cocode module for
L. In these cases we take a shortcut bypassing the uniform route we just outlined.
This is because the initial bound on r given by the columns in Table 7.2.1 is a little
too weak to pin down the structure of appropriate 2-locals, without a much more
detailed analysis of elementary subgroups of M and their fixed points on V, andwe wish to avoid that analysis.
In fact we will be able to eliminate these configurations, which correspond to
the shadows of the Fischer groups, not by directly constructing a local subgroupthat is not strongly quasithin, but instead by the use of techniques of pushing up
from sections C.2, C.3, and C.4. These results implicitly rule out a number of locals
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