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.
DEPARTMENT OF MATHEMATICS, RUTGERS UNIVERSITY, NE W BRUNSWICK,
NE W JERSEY
DEPARTMENT OF MATHEMATICS, T H E OHIO STATE UNIVERSITY, COLUM-
BUS, OHIO 43210