In Volume 4 of our series, we began the proof of the Classification Theorem
by establishing certain uniqueness and pre-uniqueness results, as formulated in our
Uniqueness Grid in Volume 1. Although Volume 4 does not contain all of the
uniqueness results we shall eventually need, we now move in this volume to begin
the proof of a major theorem from the Classification Grid, namely Theorem 67.
By the measure of the number of groups whose identification it achieves, Theo-
rem C7 is the leading theorem in the series, and it is for this reason that we present
it before Theorems 62-66? which logically precede it. Indeed Theorem 67 aims at
the identification of the alternating groups An for n 13, almost all of the groups
of Lie type of twisted rank at least 3 (in odd characteristic) or 4 (in even charac-
teristic), as well as U$(q), q odd,
n 1. Thus we refer to this
analysis as the Generic Case.
Our approach to the Generic Case is to study first the centralizers of certain
semisimple elements of G (that is, elements which will turn out to be semisimple
when G is identified as a group of Lie type), obtaining sufficient data about the gen-
eralized Fitting subgroups of these centralizers to pave the way for the identification
of G. Indeed the central hypothesis of this case is that for some x £ G of prime
order p, CG(X) has ap-component K which is "generic" in the sense that K/Op(K)
belongs to the set 9p which basically consists of suitably large alternating groups
and groups of Lie type in characteristics other than p.
In its broadest outlines, this strategy goes back to the beginning of the mod-
ern era of the Classification endeavor, specifically to the work of Richard Brauer,
as announced at the International Congress of Mathematicians in Amsterdam in
1954. Brauer, his students and others, in particular P. Fong, W. J. Wong and
K. W. Phan, established a series of theorems in the 1950's and 1960's in which the
hypotheses were that the centralizer of some involution of G was precisely isomor-
phic to the centralizer of some involution in a target simple group G* of Lie type
in odd characteristic, and the conclusion in general was that G = G* (although in
one of Brauer's earliest announced results in this vein, CG{Z) = GL2(S) leads both
to G ^ PSL3(3) and to G = M
) .
In order to make use of such results in a general classification scheme, one must
first answer a question which Danny Gorenstein liked to pose: Why, in a finite sim-
ple group, don't the centralizers of involutions (and p-local subgroups in general)
have arbitrarily complicated structure? Gorenstein and John H. Walter undertook
a profound study of this question in the late 1960's, directed at semisimple elements
of order 2. Building on earlier work of Thompson and others, they developed the
Signalizer Functor Method for the analysis of finite groups containing a suitably
large elementary abelian 2-group (of order 8, at least), highlighting the fundamental
issues of balance and generation [GW3,GW4]. Gorenstein also proved the first
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