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Finite Soluble Groups : De Gruyter Expositions in Mathematics - Klaus Doerk

Finite Soluble Groups

De Gruyter Expositions in Mathematics

Hardcover Published: 1st May 1992
ISBN: 9783110128925
Number Of Pages: 901
For Ages: 22+ years old

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The 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 Ceara, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2018)
Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Botjan Gabrovek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Preface
Notes for the reader
Prerequisites--general group theory
Groups and subgroups--the rudimentsp. 1
Groups and homomorphismsp. 5
Seriesp. 7
Direct and semidirect productsp. 9
G-sets and permutation representationsp. 17
Sylow subgroupsp. 21
Commutatorsp. 22
Finite nilpotent groupsp. 25
The Frattini subgroupp. 30
Soluble groupsp. 34
Theorems of Gaschutz, Schur-Zassenhaus, and Maschkep. 38
Coprime operator groupsp. 41
Automorphism groups induced on chief factorsp. 44
Subnormal subgroupsp. 47
Primitive finite groupsp. 52
Maximal subgroups of soluble groupsp. 57
The transferp. 60
The wreath productp. 62
Subdirect and central productsp. 73
Extraspecial p-groups and their automorphism groupsp. 77
Automorphisms of abelian groupsp. 83
Prerequisites--representation theory
Tensor productsp. 90
Projective and injective modulesp. 95
Modules and representations of K-algebrasp. 101
The structure of a group algebrap. 111
Changing the field of a representationp. 120
Induced modulesp. 129
Clifford's theoremsp. 139
Homogeneous modulesp. 153
Representations of abelian and extraspecial groupsp. 157
Faithful and simple modulesp. 172
Modules with special propertiesp. 182
Group constructions using modulesp. 190
Introduction to soluble groups
Preparations for the [actual symbols not reproducible]-theorem of Burnsidep. 204
The proof of Burnside's [actual symbols not reproducible]-theoremp. 210
Hall subgroupsp. 216
Hall systems of a finite soluble groupp. 220
System normalizersp. 235
Pronormal subgroupsp. 241
Normally embedded subgroupsp. 250
Classes of groups and closure operations
Classes of groups and closure operationsp. 262
Some special classes defined by closure propertiesp. 271
Projectors and Schunck classes
A historical introductionp. 279
Schunck classes and boundariesp. 282
Projectors and covering subgroupsp. 288
Examplesp. 302
Locally-defined Schunck classes and other constructionsp. 321
Projectors in subgroupsp. 328
The theory of formations
Examples and basic resultsp. 333
Connections between Schunck classes and formationsp. 344
Local formationsp. 356
The theorem of Lubeseder and the theorem of Baerp. 366
Projectors and local formationsp. 375
Theorems about f-hypercentral actionp. 386
Normalizers
Normalizers in generalp. 394
Normalizers associated with a formation functionp. 396
F-normalizersp. 400
Connections between normalizers and projectorsp. 408
Precursive subgroupsp. 414
Further theory of Schunck classes
Strong containment and the lattice of Schunck classesp. 426
Complementation in the latticep. 440
D-classesp. 453
Schunck classes with normally embedded projectorsp. 461
Schunck classes with permutable and CAP projectorsp. 471
Further theory of formations
The formation generated by a single groupp. 479
Supersoluble groups and chief factor rankp. 483
Primitive saturated formationsp. 497
The saturation of a formationp. 502
Strong containment for saturated formationsp. 509
Extreme classesp. 516
Saturated formations with the cover-avoidance propertyp. 528
Injectors and Fitting sets
Historical introductionp. 535
Injectors and Fitting setsp. 537
Normally embedded subgroups are injectorsp. 548
Fischer sets and Fischer subgroupsp. 554
Fitting classes--examples and properties related to injectors
Fundamental factsp. 563
Constructions and examplesp. 574
Fischer classes, normally embedded, and permutable Fitting classesp. 600
Dominance and some characterizations of injectorsp. 617
Dark's construction--the themep. 630
Dark's construction--variationsp. 647
Fitting classes--the Lockett section
The definition and basic properties of the Lockett sectionp. 677
Fitting classes and wreath productsp. 697
Normal Fitting classesp. 704
The Lausch groupp. 720
Examples of Fitting pairs and Berger's theoremp. 737
The Lockett conjecturep. 761
Fitting classes--their behaviour as classes of groups
Fitting formationsp. 775
Metanilpotent Fitting classes with additional closure propertiesp. 783
Further theory of metanilpotent Fitting classesp. 799
Fitting class boundaries Ip. 806
Fitting class boundaries IIp. 816
Frattini duals and Fitting classesp. 824
Appendix [alpha]. A theorem of Oates and Powellp. 833
Appendix [beta]. Frattini extensionsp. 846
Bibliographyp. 855
List of Symbolsp. 871
Index of Subjectsp. 873
Index of Namesp. 889
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783110128925
ISBN-10: 3110128926
Series: De Gruyter Expositions in Mathematics
Audience: Professional
For Ages: 22+ years old
Format: Hardcover
Language: English
Number Of Pages: 901
Published: 1st May 1992
Publisher: De Gruyter
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 4.45
Weight (kg): 1.79

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