Mathematical Surveys and Monographs 1994; 165 pp; softcover Volume: 40 ISBN10: 0821809601 ISBN13: 9780821809600 List Price: US$45 Individual Members: US$27 Institutional Members: US$36 Order Code: SURV/40.1.S
 The classification of the finite simple groups is one of the major feats of contemporary mathematical research, but its proof has never been completely extricated from the journal literature in which it first appeared. This book serves as an introduction to a series devoted to organizing and simplifying the proof. The purpose of the series is to present as direct and coherent a proof as is possible with existing techniques. This first volume, which sets up the structure for the entire series, begins with largely informal discussions of the relationship between the Classification Theorem and the general structure of finite groups, as well as the general strategy to be followed in the series and a comparison with the original proof. Also listed are background results from the literature that will be used in subsequent volumes. Next, the authors formally present the structure of the proof and the plan for the series of volumes in the form of two grids, giving the main case division of the proof as well as the principal milestones in the analysis of each case. Thumbnail sketches are given of the ten or so principal methods underlying the proof. Much of the book is written in an expository style accessible to nonspecialists. Readership The material here ranges from exposition suitable to a first or secondyear graduate student to more technical portions suitable for specialists. Reviews "This new, secondgeneration proof is a great improvement while still being mostly based on the methods developed for the first proof ... The entire organization of the proof is given in great detail. The wait for this project to reach this stage has been worth it."  Mathematical Reviews "Written in extremely careful and clear style ... readable, and will repay those who work through it by giving them a detailed picture of the strategy of the whole proof."  Bulletin of the London Mathematical Society Table of Contents Part I. Chapter 1, Overview  Introduction to the series
 The finite simple groups
 The structure of finite groups
 Classifying simple groups
 The background results
 Sketch of the simplified proof
 Additional comments
Part I. Chapter 2, Outline of Proof  Introduction
 The grids
 The uniqueness grid
 The classification grid: Generic and special simple groups
 The classification grid: The stages of the proof
 Principal techniques of the proof
 Notational conventions
 Background references
 Expository references
 Glossary
 Index
