the several beautiful expositions of this theory already existing: the pioneering
Seminaire Chevalley as well as books by A. Borel, J. E. Humphreys and T. A.
Springer [Ch3, Borl, Hum2, Spl].
The next three chapters develop, with proofs, the consequences for the local
structure of the finite groups of Lie type. The main theme of these chapters is
computation, but we also include proofs of the Curtis-Tits theorem and related
recognition theorems. The bulk of the computation in these chapters is not new,
the principles having been set forth in general and executed in detail by A. Borel,
R. W. Carter, C. Chevalley, C. W. Curtis, N. Iwahori, R. Ree, T. A. Springer, and
R. Steinberg, as well as H. Azad, N. Burgoyne and C. Williamson, B. Cooperstein
and G. Mason, D. I. Deriziotis, M. W. Liebeck, G. M. Seitz and many others drawn
to the beautiful detail of these groups.
Rounding out the picture of the simple DC-groups is a chapter on the alter-
nating and sporadic groups. The basic properties of the sporadic simple groups
which we need—primarily the centralizers of automorphisms of prime order—have
been assumed as Background Results and are transcribed for the most part from
[GLl] and M. Aschbacher's Sporadic Groups [A2]. Additional consequences of
these are derived in this volume. Furthermore, we have taken a couple of well-
established facts from the Atlas of Finite Groups [CCNPW1] and Aschbacher's
recent 3-Transposition Groups [A19]; these references are all clearly labelled, and
our inclusion of these two books as Background References is limited to these par-
ticular citations in Section 5.3.
Our sixth chapter discusses Schur multipliers and investigates some minute
details about certain exceptional covering groups of simple groups which will be of
later use, for one thing because of their connection with certain sporadic groups.
As discussed in our first volume, it is only by including [GLl] that we can
keep our list of Background References as short as it is. There are in fact precisely
two important cases where the results quoted from [GLl] depend on a number
of further references to the literature: the local properties of sporadic groups not
found in [A2] and the structure of the Schur multipliers of all the finite simple
groups, beyond those covered by the general theory for groups of Lie type found
in [Stl] and for alternating groups in, say, [Sul]. Our point of view is that it is
a separate task to assemble careful and complete expositions of these two sizable
theories, as indeed Aschbacher has been doing in the case of sporadic groups with
his two books.
As promised in our preceding volume, we apply the basic structural results
developed in the first six chapters of this volume to verify in our final chapter that
every DC-group has properties (5), (5P), (Gp), {Bp), (Cp) and (Mp) for all primes p,
properties required for some basic general results such as Lv -balance. We also set
the stage for the signalizer functor method by establishing several basic generation
and balance theorems for DC-groups, many due originally to Gary Seitz. In this
chapter we proceed under the assumption that all the simple sections of the simple
group K under inspection are known simple groups. Our investigation of maximal
subgroups of small simple groups is also made with the benefit of this assumption.
In the later applications of these results K will be a proper section of our DC-proper
simple group G and so the hypothesis will be justified. Certainly it could be avoided
and "clean" proofs given (as they were originally) for many, if not all, of the results,
but often only at a considerable cost in effort and space. Indeed this is another way
in which we have been able to limit the number of Background References.
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