Preface

The term "general" in our title requires some explanation; this is not a general

text in the usual senses of the word. We mean that, by and large, our development

is part of abstract finite group theory rather than the theory of 3C-groups; that is,

our theorems are about all finite groups (or all finite simple groups) rather than

specific simple groups, and our proofs are by general arguments rather than by case-

by-case analysis of possible composition factors. In fact, however, this statement

cannot quite be made categorically, and we discuss in the introductory sections how

X-groups impinge on our exposition.

Since our overriding purpose is to lay the groundwork for the forthcoming

analysis of finite simple groups, our choice of topics and theorems has been entirely

dictated by what we shall need in future chapters. One consequence of this is that

certain subtheories — such as representation theory, permutation groups, p-groups

and solvable groups, to name four — are treated either very briefly or from a narrow

perspective. Another is that we make room for some quite specialized topics which

will be necessary. On the other hand, our organization focuses attention on topics

of importance for the analysis of simple groups.

We have chosen as the most natural starting place the theory of components,

layers and the generalized Fitting subgroup, a subject largely developed since

Gorenstein's basic text of 1968 and central to today's outlook on the structure

of finite groups. This has the effect of plunging the reader abruptly into some of

the most important but possibly unfamiliar material in the book. Those readers

wishing to begin with a review of more familiar topics might choose to read Sections

9 through 12 before embarking on Section B.

Although the book is definitely not self-contained, relying for proofs on the

standard texts as well as a few further Background References, our intention has

been to give readable treatments of the various topics, with references for proofs

freely made to the supporting texts.

We remain grateful to all the people whose help we acknowledged in the first

book in this series; again we extend our thanks. In addition we thank Bil Gonzalez

and Christine Sylanov for their assistance with word-processing, and Sergei Gelfand

for his sound advice. Most importantly, we offer a thought of gratitude to the

memory of the brilliant and inimitable Danny Gorenstein.

July, 1995

RICHARD LYONS

DEPARTMENT OF MATHEMATICS, RUTGERS UNIVERSITY, NE W BRUNSWICK,

NE W JERSEY

08903

RONALD SOLOMON

DEPARTMENT OF MATHEMATICS, T H E OHIO STATE UNIVERSITY, COLUM-

BUS, OHIO 43210

xi