00323 - The Classification of the Finite Simple Groups [Gorenstein-Lyons-Solomon].pdf

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Mathematical
Surveys
and
Monographs
Volume 40.1
The Classification
of the Finite
Simple Groups
Daniel Gorenstein
Richard Lyons
Ronald Solomon
American Mathematical Society
Providence, Rhode Island
The authors were supported in part by NSF grant #DMS 89-03124, by DIMACS, an
NSF Science and Technology Center funded under contract STC-88-09648, and by NSA
grant #MDA-904-91-H-0043.
2000 Mathematics Subject Classification.
Primary 20D05, 20E32.
ABSTRACT. This is the first monograph in a series devoted to a revised proof of the classification
of the finite simple groups.
Library of
Congress
Cataloging-in-Publication Data
Gorenstein, Daniel.
The classification of the finite simple groups / Daniel Gorenstein, Richard Lyons, Ronald
Solomon.
p. cm. - (Mathematical surveys and monographs, v. 40, number 1-)
Includes bibliographical references and index.
ISBN 0-8218-0334-4
1.
Finite simple groups. I. Lyons, Richard, 1945-. Il. Solomon, Ronald. IlI. Title. IV.
Series: Mathematical surveys and monographs; no. 40, pt. 1-.
QA177.G67 1994
512'.2-dc20
94-23001
CIP
Copying and reprinting.
Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Assistant to the Publisher, American Mathematical Society,
P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made bye-mail to
reprint-permission@ams.org.
©
1994 by the American Mathematical Society. All rights reserved.
Reprinted with corrections, 2000.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
established to ensure permanence and durability.
Visit the AMS home page at URL: http://www . ams. org/
109876 5432
05 04 03 02 01 00
@)
The paper used in this book is acid-free and falls within the guidelines
To the memory of
Pearl Solomon (1918-1978)
and
Marvin Lyons (1903-1975)
Contents
Preface
Preface to the Second Printing
Part I, Chapter 1: Overview
Introduction to the Series
A. The Finite Simple Groups
1. Simple groups
2. X-groups
B. The Structure of Finite Groups
3. The Jordan-Holder theorem and simple groups
4. The generalized Fitting subgroup and quasisimple groups
5. p'-cores
and p-components
6. The embedding of
p'
-cores and p-components
7. Terminal and p-terminal p-components
8. p-constrained and p-solvable groups
C. Classifying Simple Groups
9. Internal analysis: targeting local structure
10. Internal analysis: passing from global to local information
11. Identifying simple groups
12. A capsule summary of this series
13. The existing classification proof
14. Simplifying the classification theorem
D. The Background Results
15. Foundational material
16. The initial assumptions
17. Background Results: basic material
18. Background Results: revised portions of the proof and
selected papers
E. Sketch of the Simplified Proof
19. Centralizers of semisimple elements
20. Uniqueness subgroups
21. The sets
'cp
(G)
and groups of even type
22. Generic simple groups and neighborhoods
23. The main case division
24. Special simple groups
25. Stages of the proof
26. Generic simple groups
27. The identification of
G
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