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Finite Groups
 
Daniel Gorenstein
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-8049-3
Product Code:  CHEL/301.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-8057-8
Product Code:  CHEL/301.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-8049-3
eBook: ISBN:  978-1-4704-8057-8
Product Code:  CHEL/301.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
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Finite Groups
Daniel Gorenstein
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-8049-3
Product Code:  CHEL/301.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-8057-8
Product Code:  CHEL/301.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-8049-3
eBook ISBN:  978-1-4704-8057-8
Product Code:  CHEL/301.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 301; 1968; 519 pp
    MSC: Primary 20

    ...it is required reading for anyone who wishes to study the subject.

    —Mathematical Reviews

    The theory of finite simple groups enjoyed a period of spectacular activity in the 1950s and 1960s. The first edition of Gorenstein's book was published in 1968, at the time of some of the first major classification results. The second edition was published in 1980, when it was clear that the classification was understood and the proof was within reach. Gorenstein's treatment of the subject proved prescient, as many of the developments between the two editions could be seen as continuations of the material in the book. Even now, the book remains one of the best sources for an introduction to finite groups and the classification of the simple groups. Gorenstein's insight provides a guiding light through the many pages that have been dedicated to the proof.
    Readership

    Undergraduates, graduate students, and research mathematicians interested in finite groups and the classification of the simple groups.
  • Table of Contents
     
     
    • Front Cover
    • CONTENTS
    • PREFACE TO THE SECOND EDITION
    • PREFACE TO THE FIRST EDITION
    • PART I METHODS
    • CHAPTER 1 PRELIMINARIES
    • 1. NOTATION AND TERMINOLOGY
    • 2. ASSUMED RESULTS
    • 3. RELATED ELEMENTARY RESULTS
    • EXERCISES
    • CHAPTER 2 SOME BASIC TOPICS
    • 1. CHARACTERISTIC SUBGROUPS
    • 2. ELEMENTARY PROPERTIES OF COMMUTATORS
    • 3. NILPOTENT GROUPS
    • 4. SOLVABLE GROUPS
    • 5. SEMIDIRECT AND CENTRAL PRODUCTS
    • 6. AUTOMORPHISMS AS LINEAR TRANSFORMATIONS
    • 7. TRANSITIVE AND DOUBLY TRANSITIVE PERMUTATION GROUPS
    • 8. THE TWO-DIMENSIONAL LINEAR AND PROJECTIVE GROUPS
    • EXERCISES
    • CHAPTER 3 REPRESENTATIONS OF GROUPS
    • 1. BASIC CONCEPTS
    • 2. REPRESENTATIONS OF ABELIAN GROUPS
    • 3. COMPLETE REDUCIBILITY
    • 4. CLIFFORD'S THEOREM
    • 5. G-HOMOMORPHISMS
    • 6. IRREDUCIBLE REPRESENTATIONS AND GROUP ALGEBRAS
    • 7. REPRESENTATIONS OF DIRECT AND CENTRAL PRODUCTS
    • 8. p-STABLE REPRESENTATIONS
    • EXERCISES
    • CHAPTER 4 CHARACTER THEORY
    • 1. BASIC PROPERTIES
    • 2. THE ORTHOGONALITY RELATIONS
    • 3. SOME APPLICATIONS
    • 4. INDUCED CHARACTERS AND TRIVIAL INTERSECTION SETS
    • 5. FROBENIUS GROUPS
    • 6. COHERENCE
    • 7. BRAUER'S CHARACTERIZATION OF CHARACTERS
    • EXERCISES
    • CHAPTER 5 GROUPS OF PRIME POWER ORDER
    • 1. THE FRATTINI SUBGROUP
    • 2. p'-AUTOMORPHISMS OF ABELIAN p-GROUPS
    • 3. p'-AUTOMORPHISMS OF p-GROUPS
    • 4. p-GROUPS OF SMALL DEPTH
    • 5. EXTRA-SPECIAL p-GROUPS
    • 6. THE ASSOCIATED LIE RING
    • EXERCISES
    • CHAPTER 6 SOLVABLE AND n-SOLVABLE GROUPS
    • 1. THE FITTING AND FRATTINI SUBGROUPS
    • 2. THE SCHUR-ZASSENHAUS THEOREM
    • 3. π-SEPARABLE AND π-SOLVABLE GROUPS
    • 4. SOLVABLE GROUPS
    • 5. p-STABILITY IN p-SOLVABLE GROUPS
    • EXERCISES
    • CHAPTER 7 FUSION, TRANSFER, AND p-FACTOR GROUPS
    • 1. LOCAL FUSION
    • 2. ALPERIN'S THEOREM
    • 3. TRANSFER AND THE FOCAL SUBGROUP
    • 4. THEOREMS OF BURNSIDE, FROBENIUS, AND GRUN
    • 5. WEAK CLOSURE AND p-NORMALITY
    • 6. ELEMENTARY APPLICATIONS
    • 7. GROUPS WITH DIHEDRAL SYLOW 2-SUBGROUPS
    • EXERCISES
    • CHAPTER 8 p-CONSTRAINED AND p-STABLE GROUPS
    • 1. p-CONSTRAINT AND p-STABILITY
    • 2. GLAUBERMAN'S THEOREM
    • 3. THE GLAUBERMAN-THOMPSON NORMAL p-COMPLEMENT THEOREM
    • 4. GROUPS WITH SUBGROUPS OF GLAUBERMAN TYPE
    • 5. THE THOMPSON TRANSITIVITY THEOREM
    • 6. THE MAXIMAL SUBGROUP THEOREM
    • EXERCISES
    • CHAPTER 9 GROUPS OF EVEN ORDER
    • 1. ELEMENTARY PROPERTIES OF INVOLUTIONS
    • 2. THE FEIT-SUZUKI-THOMPSON THEOREMS
    • 3. TWO APPLICATIONS
    • 4. GROUP ORDER FORMULAS
    • EXERCISES
    • PART II APPLICATIONS
    • CHAPTER 10 FIXED-POINT-FREE AUTOMORPHISMS
    • 1. ELEMENTARY PROPERTIES
    • 2. FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER
    • 3. FROBENIUS GROUPS AND GROUPS WITH NILPOTENT MAXIMAL SUBGROUPS
    • 4. FIXED-POINT-FREE AUTOMORPHISMS OF ORDER 4
    • 5. FIXED-POINT-FREE FOUR-GROUPS OF AUTOMORPHISMS
    • EXERCISES
    • CHAPTER 11 THE HALL-HIGMAN THEOREM
    • 1. STATEMENT AND INITIAL REDUCTIONS
    • 2. THE EXTRA-SPECIAL CASE
    • EXERCISES
    • CHAPTER 12 GROUPS WITH GENERALIZED QUATERNION SYLOW 2-SUBGROUPS
    • CHAPTER 13 ZASSENHAUS GROUPS
    • 1. ELEMENTARY PROPERTIES
    • 2. FEIT'S THEOREM
    • 3. CLASSIFICATION OF CERTAIN ZASSENHAUS GROUPS
    • CHAPTER 14 GROUPS IN WHICH CENTRALIZERS ARE NILPOTENT
    • 1. BASIC PROPERTIES OF CN-GROUPS
    • 2. CN-GROUPS OF ODD ORDER
    • 3. SOLVABILITY OF CN-GROUPS OF ODD ORDER
    • 4. CN-GROUPS WITH ABELIAN SYLOW 2-SUBGROUPS
    • CHAPTER 15 GROUPS WITH SELF-CENTRALIZING SYLOW 2-SUBGROUPS OF ORDER 4
    • 1. SOME PROPERTIES OF L2(q)
    • 2. STATEMENT OF THE THEOREM AND INITIAL REDUCTION
    • 3. THE STRUCTURE OF THE CENTRALIZER OF AN INVOLUTION
    • 4. THE BRAUER-SUZUKI-WALL THEOREM
    • EXERCISES
    • PART III GENERAL CLASSIFICATION PROBLEMS
    • CHAPTER 16 SIMPLE GROUPS OF LOW RANK
    • 1. GENERAL METHODS AND OBJECTIVES
    • 2. GROUPS OF ODD ORDER
    • 3. GROUPS WITH DIHEDRAL SYLOW 2-SUBGROUPS
    • 4. C-GROUPS
    • 5. N-GROUPS
    • 6. GROUPS WITH ABELIAN SYLOW 2-SUBGROUPS
    • 7. OTHER CLASSIFICATION THEOREMS
    • ADDENDUM TO PAGE 479
    • CHAPTER 17 THE KNOWN SIMPLE GROUPS
    • 1. The Known Simple Groups
    • BlBLIOGRAPHY
    • LIST OF SYMBOLS
    • INDEX
    • Back Cover
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
AMS Chelsea Publishing
Volume: 301; 1968; 519 pp
MSC: Primary 20

...it is required reading for anyone who wishes to study the subject.

—Mathematical Reviews

The theory of finite simple groups enjoyed a period of spectacular activity in the 1950s and 1960s. The first edition of Gorenstein's book was published in 1968, at the time of some of the first major classification results. The second edition was published in 1980, when it was clear that the classification was understood and the proof was within reach. Gorenstein's treatment of the subject proved prescient, as many of the developments between the two editions could be seen as continuations of the material in the book. Even now, the book remains one of the best sources for an introduction to finite groups and the classification of the simple groups. Gorenstein's insight provides a guiding light through the many pages that have been dedicated to the proof.
Readership

Undergraduates, graduate students, and research mathematicians interested in finite groups and the classification of the simple groups.
  • Front Cover
  • CONTENTS
  • PREFACE TO THE SECOND EDITION
  • PREFACE TO THE FIRST EDITION
  • PART I METHODS
  • CHAPTER 1 PRELIMINARIES
  • 1. NOTATION AND TERMINOLOGY
  • 2. ASSUMED RESULTS
  • 3. RELATED ELEMENTARY RESULTS
  • EXERCISES
  • CHAPTER 2 SOME BASIC TOPICS
  • 1. CHARACTERISTIC SUBGROUPS
  • 2. ELEMENTARY PROPERTIES OF COMMUTATORS
  • 3. NILPOTENT GROUPS
  • 4. SOLVABLE GROUPS
  • 5. SEMIDIRECT AND CENTRAL PRODUCTS
  • 6. AUTOMORPHISMS AS LINEAR TRANSFORMATIONS
  • 7. TRANSITIVE AND DOUBLY TRANSITIVE PERMUTATION GROUPS
  • 8. THE TWO-DIMENSIONAL LINEAR AND PROJECTIVE GROUPS
  • EXERCISES
  • CHAPTER 3 REPRESENTATIONS OF GROUPS
  • 1. BASIC CONCEPTS
  • 2. REPRESENTATIONS OF ABELIAN GROUPS
  • 3. COMPLETE REDUCIBILITY
  • 4. CLIFFORD'S THEOREM
  • 5. G-HOMOMORPHISMS
  • 6. IRREDUCIBLE REPRESENTATIONS AND GROUP ALGEBRAS
  • 7. REPRESENTATIONS OF DIRECT AND CENTRAL PRODUCTS
  • 8. p-STABLE REPRESENTATIONS
  • EXERCISES
  • CHAPTER 4 CHARACTER THEORY
  • 1. BASIC PROPERTIES
  • 2. THE ORTHOGONALITY RELATIONS
  • 3. SOME APPLICATIONS
  • 4. INDUCED CHARACTERS AND TRIVIAL INTERSECTION SETS
  • 5. FROBENIUS GROUPS
  • 6. COHERENCE
  • 7. BRAUER'S CHARACTERIZATION OF CHARACTERS
  • EXERCISES
  • CHAPTER 5 GROUPS OF PRIME POWER ORDER
  • 1. THE FRATTINI SUBGROUP
  • 2. p'-AUTOMORPHISMS OF ABELIAN p-GROUPS
  • 3. p'-AUTOMORPHISMS OF p-GROUPS
  • 4. p-GROUPS OF SMALL DEPTH
  • 5. EXTRA-SPECIAL p-GROUPS
  • 6. THE ASSOCIATED LIE RING
  • EXERCISES
  • CHAPTER 6 SOLVABLE AND n-SOLVABLE GROUPS
  • 1. THE FITTING AND FRATTINI SUBGROUPS
  • 2. THE SCHUR-ZASSENHAUS THEOREM
  • 3. π-SEPARABLE AND π-SOLVABLE GROUPS
  • 4. SOLVABLE GROUPS
  • 5. p-STABILITY IN p-SOLVABLE GROUPS
  • EXERCISES
  • CHAPTER 7 FUSION, TRANSFER, AND p-FACTOR GROUPS
  • 1. LOCAL FUSION
  • 2. ALPERIN'S THEOREM
  • 3. TRANSFER AND THE FOCAL SUBGROUP
  • 4. THEOREMS OF BURNSIDE, FROBENIUS, AND GRUN
  • 5. WEAK CLOSURE AND p-NORMALITY
  • 6. ELEMENTARY APPLICATIONS
  • 7. GROUPS WITH DIHEDRAL SYLOW 2-SUBGROUPS
  • EXERCISES
  • CHAPTER 8 p-CONSTRAINED AND p-STABLE GROUPS
  • 1. p-CONSTRAINT AND p-STABILITY
  • 2. GLAUBERMAN'S THEOREM
  • 3. THE GLAUBERMAN-THOMPSON NORMAL p-COMPLEMENT THEOREM
  • 4. GROUPS WITH SUBGROUPS OF GLAUBERMAN TYPE
  • 5. THE THOMPSON TRANSITIVITY THEOREM
  • 6. THE MAXIMAL SUBGROUP THEOREM
  • EXERCISES
  • CHAPTER 9 GROUPS OF EVEN ORDER
  • 1. ELEMENTARY PROPERTIES OF INVOLUTIONS
  • 2. THE FEIT-SUZUKI-THOMPSON THEOREMS
  • 3. TWO APPLICATIONS
  • 4. GROUP ORDER FORMULAS
  • EXERCISES
  • PART II APPLICATIONS
  • CHAPTER 10 FIXED-POINT-FREE AUTOMORPHISMS
  • 1. ELEMENTARY PROPERTIES
  • 2. FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER
  • 3. FROBENIUS GROUPS AND GROUPS WITH NILPOTENT MAXIMAL SUBGROUPS
  • 4. FIXED-POINT-FREE AUTOMORPHISMS OF ORDER 4
  • 5. FIXED-POINT-FREE FOUR-GROUPS OF AUTOMORPHISMS
  • EXERCISES
  • CHAPTER 11 THE HALL-HIGMAN THEOREM
  • 1. STATEMENT AND INITIAL REDUCTIONS
  • 2. THE EXTRA-SPECIAL CASE
  • EXERCISES
  • CHAPTER 12 GROUPS WITH GENERALIZED QUATERNION SYLOW 2-SUBGROUPS
  • CHAPTER 13 ZASSENHAUS GROUPS
  • 1. ELEMENTARY PROPERTIES
  • 2. FEIT'S THEOREM
  • 3. CLASSIFICATION OF CERTAIN ZASSENHAUS GROUPS
  • CHAPTER 14 GROUPS IN WHICH CENTRALIZERS ARE NILPOTENT
  • 1. BASIC PROPERTIES OF CN-GROUPS
  • 2. CN-GROUPS OF ODD ORDER
  • 3. SOLVABILITY OF CN-GROUPS OF ODD ORDER
  • 4. CN-GROUPS WITH ABELIAN SYLOW 2-SUBGROUPS
  • CHAPTER 15 GROUPS WITH SELF-CENTRALIZING SYLOW 2-SUBGROUPS OF ORDER 4
  • 1. SOME PROPERTIES OF L2(q)
  • 2. STATEMENT OF THE THEOREM AND INITIAL REDUCTION
  • 3. THE STRUCTURE OF THE CENTRALIZER OF AN INVOLUTION
  • 4. THE BRAUER-SUZUKI-WALL THEOREM
  • EXERCISES
  • PART III GENERAL CLASSIFICATION PROBLEMS
  • CHAPTER 16 SIMPLE GROUPS OF LOW RANK
  • 1. GENERAL METHODS AND OBJECTIVES
  • 2. GROUPS OF ODD ORDER
  • 3. GROUPS WITH DIHEDRAL SYLOW 2-SUBGROUPS
  • 4. C-GROUPS
  • 5. N-GROUPS
  • 6. GROUPS WITH ABELIAN SYLOW 2-SUBGROUPS
  • 7. OTHER CLASSIFICATION THEOREMS
  • ADDENDUM TO PAGE 479
  • CHAPTER 17 THE KNOWN SIMPLE GROUPS
  • 1. The Known Simple Groups
  • BlBLIOGRAPHY
  • LIST OF SYMBOLS
  • INDEX
  • Back Cover
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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