Book contents
- Frontmatter
- Contents
- Preface
- Twenty-five years of Groups St Andrews Conferences
- Original Introduction
- 1 An elementary introduction to coset table methods in computational group theory
- 2 Applications of cohomology to the theory of groups
- 3 Groups with exponent four
- 4 The Schur multiplier: an elementary approach
- 5 A procedure for obtaining simplified defining relations for a subgroup
- 6 GLn and the automorphism groups of free metabelian groups and polynomial rings
- 7 Isoclinisms of group extensions and the Schur multiplicator
- 8 The maximal subgroups of the Chevalley group G2(4)
- 9 Generators and relations for the cohomology ring of Janko's first group in the first twenty one dimensions
- 10 The Burnside group of exponent 5 with two generators
- 11 The orientability of subgroups of plane groups
- 12 On groups with unbounded non-archimedean elements
- 13 An algorithm for the second derived factor group
- 14 Finiteness conditions and the word problem
- 15 Growth sequences relative to subgroups
- 16 On the centres of mapping class groups of surfaces
- 17 A glance at the early history of group rings
- 18 Units of group rings: a short survey
- 19 Subgroups of small cancellation groups: a survey
- 20 On the hopficity and related properties of some two-generator groups
- 21 The isomorphism problem and units in group rings of finite groups
- 22 On one-relator groups that are free products of two free groups with cyclic amalgamation
- 23 The algebraic structure of ℵ0-categorical groups
- 24 Abstracts
- 25 Addendum to: “An elementary introduction to coset table methods in computational group theory”
- 26 Addendum to: “Applications of cohomology to the theory of groups”
- 27 Addendum to: “Groups with exponent four”
- 28 Addendum to: “The Schur multiplier: an elementary approach”
18 - Units of group rings: a short survey
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- Contents
- Preface
- Twenty-five years of Groups St Andrews Conferences
- Original Introduction
- 1 An elementary introduction to coset table methods in computational group theory
- 2 Applications of cohomology to the theory of groups
- 3 Groups with exponent four
- 4 The Schur multiplier: an elementary approach
- 5 A procedure for obtaining simplified defining relations for a subgroup
- 6 GLn and the automorphism groups of free metabelian groups and polynomial rings
- 7 Isoclinisms of group extensions and the Schur multiplicator
- 8 The maximal subgroups of the Chevalley group G2(4)
- 9 Generators and relations for the cohomology ring of Janko's first group in the first twenty one dimensions
- 10 The Burnside group of exponent 5 with two generators
- 11 The orientability of subgroups of plane groups
- 12 On groups with unbounded non-archimedean elements
- 13 An algorithm for the second derived factor group
- 14 Finiteness conditions and the word problem
- 15 Growth sequences relative to subgroups
- 16 On the centres of mapping class groups of surfaces
- 17 A glance at the early history of group rings
- 18 Units of group rings: a short survey
- 19 Subgroups of small cancellation groups: a survey
- 20 On the hopficity and related properties of some two-generator groups
- 21 The isomorphism problem and units in group rings of finite groups
- 22 On one-relator groups that are free products of two free groups with cyclic amalgamation
- 23 The algebraic structure of ℵ0-categorical groups
- 24 Abstracts
- 25 Addendum to: “An elementary introduction to coset table methods in computational group theory”
- 26 Addendum to: “Applications of cohomology to the theory of groups”
- 27 Addendum to: “Groups with exponent four”
- 28 Addendum to: “The Schur multiplier: an elementary approach”
Summary
INTRODUCTION
Historically group rings appeared for the first time in a paper by A. Cayley which is also considered by many authors as the starting point of abstract group theory (e.g. Bourbaki or M. Kline). They were studied later by T. Molien, and G. Frobenius and earned a definitive status, in connection with group representation theory, after the work of R. Brauer and E. Noether, (regarding the history of group rings see).
In recent times the subject gained impetus after the inclusion of questions on group rings in I. Kaplansky's famous lists of problems. Other important facts to stimulate the area were the inclusion of sections on group rings in the books on ring theory by J. Lambeck and P. Ribemboim as well as the publication of the first book entirely devoted to the subject, due to D.S. Passman.
Since then several survey articles have appeared, namely those by A.E. Zaleskii and A.V. Mikhalev, D.S. Passman, K. Dennis and D. Farkas. Also new books on the subject have been published in recent years: A.A. Bovdi, I.B.S. Passi, D.S. Passman and S.K. Sehgal.
Considerable work has been done lately on the structure and group-theoretical properties of the group of units of a group ring.
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- Groups - St Andrews 1981 , pp. 281 - 297Publisher: Cambridge University PressPrint publication year: 1982
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