Structure Theorems of Unit Groups
Structure Theorems of Unit Groups

Author: Eric Jespers

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2015-11-13

Total Pages: 275

ISBN-13: 3110412756

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This two-volume graduate textbook gives a comprehensive, state-of-the-art account of describing large subgroups of the unit group of the integral group ring of a finite group and, more generally, of the unit group of an order in a finite dimensional semisimple rational algebra. Since the book is addressed to graduate students as well as young researchers, all required background on these diverse areas, both old and new, is included. Supporting problems illustrate the results and complete some of the proofs. Volume 1 contains all the details on describing generic constructions of units and the subgroup they generate. Volume 2 mainly is about structure theorems and geometric methods. Without being encyclopaedic, all main results and techniques used to achieve these results are included. Basic courses in group theory, ring theory and field theory are assumed as background.

Finitely Generated Commutative Monoids
Finitely Generated Commutative Monoids

Author: J. C. Rosales

Publisher: Nova Publishers

Published: 1999

Total Pages: 204

ISBN-13: 9781560726708

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A textbook for an undergraduate course, requiring only a knowledge of basic linear algebra. Explains how to compute presentations for finitely generated cancellative monoids, and from a presentation of a monoid, decide whether this monoid is cancellative, reduced, separative, finite, torsion free, group, affine, full, normal, etc. Of most interest to people working with semigroup theory, but also in other areas of algebra. Annotation copyrighted by Book News, Inc., Portland, OR

Methods in Ring Theory
Methods in Ring Theory

Author: Vesselin Drensky

Publisher: CRC Press

Published: 2021-02-27

Total Pages: 329

ISBN-13: 1000657353

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"Furnishes important research papers and results on group algebras and PI-algebras presented recently at the Conference on Methods in Ring Theory held in Levico Terme, Italy-familiarizing researchers with the latest topics, techniques, and methodologies encompassing contemporary algebra."

Orders and Generic Constructions of Units
Orders and Generic Constructions of Units

Author: Eric Jespers

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2015-11-13

Total Pages: 460

ISBN-13: 3110372940

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This two-volume graduate textbook gives a comprehensive, state-of-the-art account of describing large subgroups of the unit group of the integral group ring of a finite group and, more generally, of the unit group of an order in a finite dimensional semisimple rational algebra. Since the book is addressed to graduate students as well as young researchers, all required background on these diverse areas, both old and new, is included. Supporting problems illustrate the results and complete some of the proofs. Volume 1 contains all the details on describing generic constructions of units and the subgroup they generate. Volume 2 mainly is about structure theorems and geometric methods. Without being encyclopaedic, all main results and techniques used to achieve these results are included. Basic courses in group theory, ring theory and field theory are assumed as background.

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory
Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

Author: Gebhard Böckle

Publisher: Springer

Published: 2018-03-22

Total Pages: 753

ISBN-13: 3319705660

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This book presents state-of-the-art research and survey articles that highlight work done within the Priority Program SPP 1489 “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory”, which was established and generously supported by the German Research Foundation (DFG) from 2010 to 2016. The goal of the program was to substantially advance algorithmic and experimental methods in the aforementioned disciplines, to combine the different methods where necessary, and to apply them to central questions in theory and practice. Of particular concern was the further development of freely available open source computer algebra systems and their interaction in order to create powerful new computational tools that transcend the boundaries of the individual disciplines involved. The book covers a broad range of topics addressing the design and theoretical foundations, implementation and the successful application of algebraic algorithms in order to solve mathematical research problems. It offers a valuable resource for all researchers, from graduate students through established experts, who are interested in the computational aspects of algebra, geometry, and/or number theory.

A Course in Finite Group Representation Theory
A Course in Finite Group Representation Theory

Author: Peter Webb

Publisher: Cambridge University Press

Published: 2016-08-19

Total Pages: 339

ISBN-13: 1107162394

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This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.

Number Theory
Number Theory

Author: Henri Cohen

Publisher: Springer Science & Business Media

Published: 2008-10-10

Total Pages: 673

ISBN-13: 0387499237

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The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.

Complementation of Normal Subgroups
Complementation of Normal Subgroups

Author: Joseph Kirtland

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2017-09-11

Total Pages: 156

ISBN-13: 3110480212

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Starting with the Schur-Zassenhaus theorem, this monograph documents a wide variety of results concerning complementation of normal subgroups in finite groups. The contents cover a wide range of material from reduction theorems and subgroups in the derived and lower nilpotent series to abelian normal subgroups and formations. Contents Prerequisites The Schur-Zassenhaus theorem: A bit of history and motivation Abelian and minimal normal subgroups Reduction theorems Subgroups in the chief series, derived series, and lower nilpotent series Normal subgroups with abelian sylow subgroups The formation generation Groups with specific classes of subgroups complemented

Frobenius Splitting Methods in Geometry and Representation Theory
Frobenius Splitting Methods in Geometry and Representation Theory

Author: Michel Brion

Publisher: Springer Science & Business Media

Published: 2007-08-08

Total Pages: 259

ISBN-13: 0817644059

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Systematically develops the theory of Frobenius splittings and covers all its major developments. Concise, efficient exposition unfolds from basic introductory material on Frobenius splittings—definitions, properties and examples—to cutting edge research.