Finite Soluble Groups

✍ Scribed by Klaus Doerk; Trevor O. Hawkes

• Publisher: De Gruyter
• Year: 1992
• Tongue: English
• Leaves: 912
• Series: De Gruyter Expositions in Mathematics; 4
• Edition: Reprint 2011
• Category: Library

✦ Synopsis

"The exposition is open and easy to follow. Good motivation is the rule. The book will serve as an admirable text for a graduate course on group theory (several actually: one could scarcely hope to cover the whole in one course). Not the least of its attractions is the fund of examples at the end of each section. […]The reviewer could find little to criticise. […] The work will be immensely valuable to group theorists and particularly to those who work with finite soluble groups. The authors' hope, expressed in the preface, that the book might serve "as a text for postgraduate teaching, and also as a source of research ideas and techniques" is splendidly realised." Mathematical Reviews

"The authors’ hope that their book will serve as a basic reference in the subject area, as a text for postgraduate teaching and as a source of research ideas and techniques appears to rest on solid foundations. The book meets the specialist’s needs, being comprehensive by and large while concentrating on those parts of the subject where a coherent theoretical structure has emerged. […] The wealth of instructive examples and exercises drawn quite frequently from the original research papers should also be attractive to postgraduate teaching. […] The production is excellent. The authors prove: pleasant style is compatible with precision." Zentralblatt für Mathematik

✦ 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