Hardcover ISBN:  9780821853368 
Product Code:  SURV/172 
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Electronic ISBN:  9781470413996 
Product Code:  SURV/172.E 
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Book DetailsMathematical Surveys and MonographsVolume: 172; 2011; 347 ppMSC: Primary 20;
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 Lietype (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.ReadershipGraduate students and research mathematicians interested in classification of finite simple groups.

Table of Contents

Background and overview

0. Introduction

1. Overview: The classification of groups of GorensteinWalter type

2. Overview: The classification of groups of characteristic 2 type

Outline of the classification of groups of characteristic 2 type

3. $e(G)\leq 2$: The classification of quasithin groups

4. $e(G)=3$: The classification of rank 3 groups

5. $e(G)\geq 4$: The pretrichotomy and trichotomy theorems

6. The classification of groups of standard type

7. The classification of groups of $GF(2)$ type

8. The final contradiction: Eliminating the Uniqueness Case

Appendices

A. Some background material related to simple groups

B. Overview of some techniques used in the classification


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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 Lietype (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.

Background and overview

0. Introduction

1. Overview: The classification of groups of GorensteinWalter type

2. Overview: The classification of groups of characteristic 2 type

Outline of the classification of groups of characteristic 2 type

3. $e(G)\leq 2$: The classification of quasithin groups

4. $e(G)=3$: The classification of rank 3 groups

5. $e(G)\geq 4$: The pretrichotomy and trichotomy theorems

6. The classification of groups of standard type

7. The classification of groups of $GF(2)$ type

8. The final contradiction: Eliminating the Uniqueness Case

Appendices

A. Some background material related to simple groups

B. Overview of some techniques used in the classification