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This book is the ninth volume in a series whose goal is to furnish
a careful and largely self-contained proof of the classification
theorem for the finite simple groups. Having completed the
classification of the simple groups of odd type as well as the
classification of the simple groups of generic even type (modulo
uniqueness theorems to appear later), the current volume begins the
classification of the finite simple groups of special even type.
The principal result of this volume is a classification of the
groups of bicharacteristic type, i.e., of both even type and of
$p$-type for a suitable odd prime $p$. It is here that the largest
sporadic groups emerge, namely the Monster, the Baby Monster, the
largest Conway group, and the three Fischer groups, along with six
finite groups of Lie type over small fields, several of which play
a major role as subgroups or sections of these sporadic groups.
Thisseries is devoted to the publication of monographs, lecture
resp. seminar notes, and other materials arising from programs of
the OSU Mathemaical Research Institute. This includes proceedings
of conferences or workshops held at the Institute, and other
mathematical writings.
This book completes a trilogy (Numbers 5, 7, and 8) of the series
The Classification of the Finite Simple Groups treating the generic
case of the classification of the finite simple groups. In
conjunction with Numbers 4 and 6, it allows us to reach a major
milestone in our series--the completion of the proof of the
following theorem: Theorem O: Let G be a finite simple group of odd
type, all of whose proper simple sections are known simple groups.
Then either G is an alternating group or G is a finite group of Lie
type defined over a field of odd order or G is one of six sporadic
simple groups. Put another way, Theorem O asserts that any minimal
counterexample to the classification of the finite simple groups
must be of even type. The work of Aschbacher and Smith shows that a
minimal counterexample is not of quasithin even type, while this
volume shows that a minimal counterexample cannot be of generic
even type, modulo the treatment of certain intermediate
configurations of even type which will be ruled out in the next
volume of our series.
The book provides an outline and modern overview of the
classification of the finite simple groups. It primarily covers the
“even case”, where the main groups arising are Lie-type
(matrix) groups over a field of characteristic 2. The book thus
completes a project begun by Daniel Gorenstein’s 1983 book, which
outlined the classification of groups of “noncharacteristic 2
type”. However, this book provides much more. Chapter 0 is a
modern overview of the logical structure of the entire
classification. Chapter 1 is a concise but complete outline of the
“odd case” with updated references, while Chapter 2 sets the
stage for the remainder of the book with a similar outline of the
“even case”. The remaining six chapters describe in detail the
fundamental results whose union completes the proof of the
classification theorem. Several important subsidiary results are
also discussed. In addition, there is a comprehensive listing of
the large number of papers referenced from the literature.
Appendices provide a brief but valuable modern introduction to many
key ideas and techniques of the proof. Some improved arguments are
developed, along with indications of new approaches to the entire
classification—such as the second and third generation
projects—although there is no attempt to cover them
comprehensively. The work should appeal to a broad range of
mathematicians—from those who just want an overview of the main
ideas of the classification, to those who want a reader’s guide
to help navigate some of the major papers, and to those who may
wish to improve the existing proofs.
The classification of finite simple groups is a landmark result of
modern mathematics. The original proof is spread over scores of
articles by dozens of researchers. In this multivolume book, the
authors are assembling the proof with explanations and references.
It is a monumental task. The book, along with background from
sections of the previous volumes, presents critical aspects of the
classification. In four prior volumes (Surveys of Mathematical
Monographs, Volumes 40.1, 40.2, 40.3, and 40.4), the authors began
the proof of the classification theorem by establishing certain
uniqueness and preuniqueness results. In this volume, they now
begin the proof of a major theorem from the classification grid,
namely Theorem ${\mathcal C 7$. The book is suitable for graduate
students and researchers interested in group theory.
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