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The Classification of Finite Simple Groups: Groups of Characteristic 2 Type
Michael Aschbacher, California Institute of Technology, Pasadena, CA, Richard Lyons, Rutgers University, Piscataway, NJ, Stephen D. Smith, University of Illinois at Chicago, IL, and Ronald Solomon, The Ohio State University, Columbus, OH
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Mathematical Surveys and Monographs
2011; 347 pp; hardcover
Volume: 172
ISBN-10: 0-8218-5336-8
ISBN-13: 978-0-8218-5336-8
List Price: US$94 Member Price: US$75.20
Order Code: SURV/172

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.

Graduate students and research mathematicians interested in classification of finite simple groups.