An Introduction to Group Rings

by César Polcino Milies, S.K. Sehgal

Overview

Group rings play a central role in the theory of representations of groups and are very interesting algebraic objects in their own right. In their study, many branches of algebra come to a rich interplay. This book takes the reader from beginning to research level and contains many topics that, so far, were only found in papers published in scientific journals and, whenever possible, offers new proofs of known results. It also includes many historical notes and some applications.
Audience: This book will be of interest to mathematicians working in the area of group rings and it serves as an introduction of the subject to graduate students.

Product Details

ISBN-13:	9781402002380
Publisher:	Springer Netherlands
Publication date:	01/31/2002
Series:	Algebra and Applications , #1
Edition description:	2002
Pages:	371
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1 Groups.- 1.1 Basic Concepts.- 1.2 Homomorphisms and Factor Groups.- 1.3 Abelian Groups.- 1.4 Group Actions, p-groups and Sylow Subgroups.- 1.5 Solvable and Nilpotent Groups.- 1.6 FC Groups.- 1.7 Free Groups and Free Products.- 1.8 Hamiltonian Groups.- 1.9 The Hirsch Number.- 2 Rings, Modules and Algebras.- 2.1 Rings and Ideals.- 2.2 Modules and Algebras.- 2.3 Free Modules and Direct Sums.- 2.4 Finiteness Conditions.- 2.5 Semisimplicity.- 2.6 The Wedderburn-Artin Theorem.- 2.7 The Jacobson Radical.- 2.8 Rings of Algebraic Integers.- 2.9 Orders.- 2.10 Tensor Products.- 3 Group Rings.- 3.1 A Brief History.- 3.2 Basic Facts.- 3.3 Augmentation Ideals.- 3.4 Semisimplicity.- 3.5 Abelian Group Algebras.- 3.6 Some Commutative Subalgebras.- 4 A Glance at Group Representations.- 4.1 Definition and Examples.- 4.2 Representations and Modules.- 5 Group Characters.- 5.1 Basic Facts.- 5.2 Characters and Isomorphism Questions.- 6 Ideals in Group Rings.- 6.1 Ring Theoretic Formulas.- 6.2 Nilpotent Ideals.- 6.3 Nilpotent Augmentation Ideals.- 6.4 Semiprime Group Rings.- 6.5 Prime Group Rings.- 6.6 Chain Conditions in KG.- 7 Algebraic Elements.- 7.1 Introduction.- 7.2 Idempotent Elements.- 7.3 Torsion Units.- 7.4 Nilpotent Elements.- 8 Units of Group Rings.- 8.1 Introduction.- 8.2 Trivial Units.- 8.3 Finite Groups.- 8.4 Units of ZS3.- 8.5 Infinite Groups.- 8.6 Finite Generation of U(ZG).- 8.7 Central Units.- 9 The Isomorphism Problem.- 9.1 Introduction.- 9.2 The Normal Subgroup Correspondence.- 9.3 Metabelian Groups.- 9.4 Circle Groups.- 9.5 Further Results.- 9.6 The Modular Isomorphism Problem.- 10 Free Groups of Units.- 10.1 Free Groups.- 10.2 Free Groups of Units.- 10.3 Explicit Free Groups.- 10.4 Explicit Free Groups in H.- 11 Properties of the Unit Group.- 11.1 Integral Group Rings.- 11.2 Group Algebras.

Editorial Reviews

Sehgal (mathematical and statistical sciences, U. of Alberta, Canada) and Milies (mathematics and statistics, Universidade de São Paulo, Brasil) intend this text as an introduction to the general theory of group rings, which will take the reader from beginning to research level. The text requires a first-year graduate level knowledge of algebra. Coverage includes the basics of group representation theory and characters, the connections between this theory and the structure of group algebras, the isomorphism problem, the basic properties of ideals in group rings, algebraic elements in group rings, and the structure of the unit group. For mathematicians working in the area of group rings, and for use as an introductory text for graduate students. Annotation c. Book News, Inc., Portland, OR (booknews.com)