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The Classification of the Finite Simple Groups
Daniel Gorenstein, Richard Lyons, Rutgers University, New Brunswick, NJ, and Ronald M Solomon, Ohio State University, Columbus, OH

Mathematical Surveys and Monographs
1994; 165 pp; softcover
Volume: 40
ISBN-10: 0-8218-0960-1
ISBN-13: 978-0-8218-0960-0
List Price: US$45
Individual Members: US$27
Institutional Members: US$36
Order Code: SURV/40.1.S
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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.


The material here ranges from exposition suitable to a first- or second-year graduate student to more technical portions suitable for specialists.


"This new, second-generation 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
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