## Finite Soluble GroupsThe aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Cear , Fortaleza, Brasil Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) |

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### Contents

Chapter A Prerequisitesgeneral group theory 1 Groups and subgroupsthe rudiments | 1 |

Groups and homomorphisms | 5 |

Series | 7 |

Direct and semidirect products | 9 |

Gsets and permutation representations | 17 |

Sylow subgroups | 21 |

Commutators | 22 |

Finite nilpotent groups | 25 |

Local formations | 356 |

The theorem of Lubeseder and the theorem of Baer | 366 |

Projectors and local formations | 375 |

Theorems about hypercentral action | 386 |

Chapter V | 394 |

ftnormalizers | 400 |

Connections between normalizers and projectors | 408 |

Precursive subgroups | 414 |

The Frattini subgroup | 30 |

Soluble groups | 34 |

Theorems of Gaschiitz SchurZassenhaus and Maschke | 38 |

Coprime operator groups | 41 |

Automorphism groups induced on chief factors | 44 |

Subnormal subgroups | 47 |

Primitive finite groups | 52 |

Maximal subgroups of soluble groups | 57 |

The transfer | 60 |

The wreath product | 62 |

Subdirect and central products | 73 |

Extraspecial pgroups and their automorphism groups | 77 |

Automorphisms of abelian groups | 83 |

Chapter B Prerequisitesrepresentation theory 1 Tensor products | 90 |

Projective and injective modules | 95 |

Modules and representations of Kalgebras | 101 |

The structure of a group algebra | 111 |

Changing the field of a representation | 120 |

Induced modules | 129 |

Faithful and simple modules | 172 |

Modules with special properties | 182 |

Group constructions using modules | 190 |

Chapter I | 204 |

The proof of Burnsides ptheorem | 210 |

Hall subgroups | 216 |

System normalizers | 235 |

Pronormal subgroups | 241 |

Normally embedded subgroups | 250 |

Chapter II | 262 |

Some special classes defined by closure properties | 271 |

Chapter III | 279 |

Projectors and covering subgroups | 288 |

Examples | 302 |

Locallydefined Schunck classes and other constructions | 321 |

Projectors in subgroups | 328 |

Connections between Schunck classes and formations | 344 |

Chapter VI | 426 |

Complementation in the lattice | 440 |

Schunck classes with normally embedded projectors | 461 |

Schunck classes with permutable and CAP projectors | 471 |

Chapter VII | 479 |

Primitive saturated formations | 497 |

Strong containment for saturated formations | 509 |

Extreme classes | 516 |

Saturated formations with the coveravoidance property | 528 |

Chapter VIII | 535 |

Normally embedded subgroups are injectors | 548 |

Fischer sets and Fischer subgroups | 554 |

Fitting classesexamples and properties related to injectors | 563 |

Constructions and examples | 574 |

Fischer classes normally embedded and permutable Fitting classes | 600 |

Dominance and some characterizations of injectors | 617 |

Darks constructionthe theme | 630 |

Darks constructionvariations | 647 |

Chapter X | 677 |

Fitting classes and wreath products | 697 |

Normal Fitting classes | 704 |

The Lausch group | 720 |

Examples of Fitting pairs and Bergers theorem | 737 |

The Lockett conjecture | 761 |

Chapter XI | 775 |

Metanilpotent Fitting classes with additional closure properties | 783 |

Further theory of metanilpotent Fitting classes | 799 |

Fitting class boundaries I | 806 |

Fitting class boundaries II | 816 |

Frattini duals and Fitting classes | 824 |

Appendix a A theorem of Gates and Powell | 833 |

Appendix 3 Frattini extensions | 846 |

855 | |

871 | |

889 | |

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### References to this book

The Theory of Finite Groups: An Introduction Hans Kurzweil,Bernd Stellmacher No preview available - 2003 |