Finite Soluble Groups

Klaus Doerk, Trevor O. Hawkes

• Publisher: De Gruyter
• Year: 2011
• ISBN:
  9783110870138
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Table of contents

• Preface
• Notes for the reader
• Chapter A Prerequisites — general group theory
• 1. Groups and subgroups — the rudiments
• 2. Groups and homomorphisms
• 3. Series
• 4. Direct and semidirect products
• 5. G-sets and permutation representations
• 6. Sylow subgroups
• 7. Commutators
• 8. Finite nilpotent groups
• 9. The Frattini subgroup
• 10. Soluble groups
• 11. Theorems of Gaschütz, Schur-Zassenhaus, and Maschke
• 12. Coprime operator groups
• 13. Automorphism groups induced on chief factors
• 14. Subnormal subgroups
• 15. Primitive finite groups
• 16. Maximal subgroups of soluble groups
• 17. The transfer
• 18. The wreath product
• 19. Subdirect and central products
• 20. Extraspecial p-groups and their automorphism groups
• 21. Automorphisms of abelian groups
• Chapter B Prerequisites — representation theory
• 1. Tensor products
• 2. Projective and injective modules
• 3. Modules and representations of K-algebras
• 4. The structure of a group algebra
• 5. Changing the field of a representation
• 6. Induced modules
• 7. Clifford’s theorems
• 8. Homogeneous modules
• 9. Representations of abelian and extraspecial groups
• 10. Faithful and simple modules
• 11. Modules with special properties
• 12. Group constructions using modules
• Chapter I. Introduction to soluble groups
• 1. Preparations for the paqb-theorem of Burnside
• 2. The proof of Burnside’s paqb-theorem
• 3. Hall subgroups
• 4. Hall systems of a finite soluble group
• 5. System normalizers
• 6. Pronormal subgroups
• 7. Normally embedded subgroups
• Chapter II Classes of groups and closure operations
• 1. Classes of groups and closure operations
• 2. Some special classes defined by closure properties
• Chapter III. Projectors and Schunck classes
• 1. A historical introduction
• 2. Schunck classes and boundaries
• 3. Projectors and covering subgroups
• 4. Examples
• 5. Locally-defined Schunck classes and other constructions
• 6. Projectors in subgroups
• Chapter IV. The theory of formations
• 1. Examples and basic results
• 2. Connections between Schunck classes and formations
• 3. Local formations
• 4. The theorem of Lubeseder and the theorem of Baer
• 5. Projectors and local formations
• 6. Theorems about f-hypercentral action
• Chapter V. Normalizers
• 1. Normalizers in general
• 2. Normalizers associated with a formation function
• 3. ℱ-normalizers
• 4. Connections between normalizers and projectors
• 5. Precursive subgroups
• Chapter VI. Further theory of Schunck classes
• 1. Strong containment and the lattice of Schunck classes
• 2. Complementation in the lattice
• 3. D-classes
• 4. Schunck classes with normally embedded projectors
• 5. Schunck classes with permutable and CAP projectors
• Chapter VII. Further theory of formations
• 1. The formation generated by a single group
• 2. Supersoluble groups and chief factor rank
• 3. Primitive saturated formations
• 4. The saturation of a formation
• 5. Strong containment for saturated formations
• 6. Extreme classes
• 7. Saturated formations with the cover-avoidance property
• Chapter VIII. Injectors and Fitting sets
• 1. Historical introduction
• 2. Injectors and Fitting sets
• 3. Normally embedded subgroups are injectors
• 4. Fischer sets and Fischer subgroups
• Chapter IX. Fitting classes — examples and properties related to injectors
• 1. Fundamental facts
• 2. Constructions and examples
• 3. Fischer classes, normally embedded, and permutable Fitting classes
• 4. Dominance and some characterizations of injectors
• 5. Dark’s construction — the theme
• 6. Dark’s construction — variations
• Chapter X. Fitting classes — the Lockett section
• 1. The definition and basic properties of the Lockett section
• 2. Fitting classes and wreath products
• 3. Normal Fitting classes
• 4. The Lausch group
• 5. Examples of Fitting pairs and Berger’s theorem
• 6. The Lockett conjecture
• Chapter XI. Fitting classes — their behaviour as classes of groups
• 1. Fitting formations
• 2. Metanilpotent Fitting classes with additional closure properties
• 3. Further theory of metanilpotent Fitting classes
• 4. Fitting class boundaries I
• 5. Fitting class boundaries II
• 6. Frattini duals and Fitting classes
• Appendix α. A theorem of Oates and Powell
• Appendix β. Frattini extensions
• Bibliography
• List of Symbols
• Index of Subjects
• Index of Names