In Part 3, we consider most cases where the Fundamental Setup (3.2.1) holds for
a pair L, V such that Vis not a failure of factorization module for N GL(V) (AutLo (V))
where Lo := (LT). The object of Part 3 is to eliminate all but one of the pairs
considered here: we will show that G ~ J4 when V is the cocode module for
M/V ~ M 24 , and that none of the other pairs lead to examples. However we will
also have to deal with a number of shadows whose local subgroups possess the pairs
considered in this chapter.
THEOREM Assume the Fundamental Setup (3.2.1). Then one of the follow-
ing holds:
(1) Vis an FF-module for NaL(V)(AutL 0 (V)).
(2) Vis the cocode module for M/V ~ M24 and G ~ J4.(3) Vis the orthogonal module for AutL 0 (V) ~ L2(2^2 n) ~ D4(2n), with n > 1.
(4) Conclusion (3) of 3.2.6 is satisfied. In particular L < Lo and L/0 2 (L) ~
L2(2n), Sz(2n), or L3(2).
Note that case (3) and a part of case (1) were handled earlier in Part 2; while
case (4) and' the remainder of case (1) will be handled later in Part 4 and Part 5.
In the initial chapter of Part 3, we begin to implement the outline for weak
closure arguments described in subsection E.3.3. The cases not corresponding to
shadows or J4 will then be quickly eliminated by comparing various parameters
associated to the representation of LoT on V. The remaining two chapters in Part
3 will pursue the deeper analysis required when the configurations do correspond
to shadows or J4.