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The Classification of the Finite Simple Groups, Number 4: Part II, Chapters 1-4: Uniqueness Theorems
Daniel Gorenstein, Richard Lyons, Rutgers University, New Brunswick, NJ, and Ronald Solomon, Ohio State University, Columbus, OH

Mathematical Surveys and Monographs
1999; 341 pp; hardcover
Volume: 40
ISBN-10: 0-8218-1379-X
ISBN-13: 978-0-8218-1379-9
List Price: US$87
Member Price: US$69.60
Order Code: SURV/40.4
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After three introductory volumes on the classification of the finite simple groups, (Mathematical Surveys and Monographs, Volumes 40.1, 40.2, and 40.3), the authors now start the proof of the classification theorem: They begin the analysis of a minimal counterexample \(G\) to the theorem.

Two fundamental and powerful theorems in finite group theory are examined: the Bender-Suzuki theorem on strongly embedded subgroups (for which the non-character-theoretic part of the proof is provided) and Aschbacher's Component theorem. Included are new generalizations of Aschbacher's theorem which treat components of centralizers of involutions and \(p\)-components of centralizers of elements of order \(p\) for arbitrary primes \(p\).

This book, with background from sections of the previous volumes, presents in an approachable manner critical aspects of the classification of finite simple groups.


  • Treatment of two fundamental and powerful theorems in finite group theory.
  • Proofs that are accessible and largely self-contained.
  • New results generalizing Aschbacher's Component theorem and related component uniqueness theorems.


Graduate students and research mathematicians working in group theory and generalizations.

Table of Contents

  • General lemmas
  • Strongly embedded subgroups and related conditions on involutions
  • \(p\)-component uniqueness theorems
  • Properties of \(K\)-groups
  • Background references
  • Expository references
  • Errata for number 3, Chapter I\(_A\): Almost simple \(\mathcal K\)-groups
  • Glossary
  • Index of terminology
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