## Finite Soluble GroupsAt least one year of the Wall Street journal (WSJ) on a single disk, updated monthly, and subject to Boolean search (excluding Reuters no great loss and the ads, the digest of earnings and the dividends tables, futures prices, and stock tables and other free-standing tabular data). Includes some 40, |

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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 | 409 |

Chapter VI | 426 |

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 |

Complementation in the lattice | 440 |

Schunck classes with normally embedded projectors | 461 |

Schunck classes with permutable and CAP projectors | 471 |

Chapter VII | 477 |

Supersoluble groups and chief factor rank | 485 |

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 | 676 |

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 | |

### Common terms and phrases

abelian group Assertion assume automorphism belongs Carter subgroup central chief factors chief series class of groups closure operation complement composition factor conclude Condition conjugacy class conjugate consequently contradiction Corollary cyclic deduce defined direct product element elementary abelian epimorphism example exists factor of G faithful finite group finite soluble group Fitting class Fitting class g Fitting set Frattini G containing g e G Gaschiitz group G group of order Hall system hence homomorphism hypothesis implies induction injector of G irreducible isomorphic Lemma Let F Let G Let H LF(f maximal subgroup minimal normal subgroup minimal order nilpotent non-abelian non-empty non-trivial normally embedded notation obviously p-chief p-group particular permutable primitive group pronormal Proof Proposition prove Q-closed quotient satisfies saturated formation Schunck class semidirect product simple module Soc(G subgroup of G submodule subnormal subgroup suppose Sylow p-subgroup Theorem wreath product