xiv PREFACE

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 needâ€”primarily the centralizers of automorphisms of prime orderâ€”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.