# Group Rings and Class Groups

The first part of the book centers around the isomorphism problem for finite groups; i.e. which properties of the finite group G can be determined by the integral group ring ZZG ?

Author: K.W. Roggenkamp

Publisher: Birkhäuser

ISBN: 9783034886116

Category: Mathematics

Page: 210

View: 674

The first part of the book centers around the isomorphism problem for finite groups; i.e. which properties of the finite group G can be determined by the integral group ring ZZG ? The authors have tried to present the results more or less selfcontained and in as much generality as possible concerning the ring of coefficients. In the first section, the class sum correspondence and some related results are derived. This part is the proof of the subgroup rigidity theorem (Scott - Roggenkamp; Weiss) which says that a finite subgroup of the p-adic integral group ring of a finite p-group is conjugate to a subgroup of the finite group. A counterexample to the conjecture of Zassenhaus that group basis are rationally conjugate, is presented in the semilocal situation (Scott - Roggenkamp). To this end, an extended version of Clifford theory for p-adic integral group rings is presented. Moreover, several examples are given to demonstrate the complexity of the isomorphism problem. The second part of the book is concerned with various aspects of the structure of rings of integers as Galois modules. It begins with a brief overview of major results in the area; thereafter the majority of the text focuses on the use of the theory of Hopf algebras. It begins with a thorough and detailed treatment of the required foundational material and concludes with new and interesting applications to cyclotomic theory and to elliptic curves with complex multiplication. Examples are used throughout both for motivation, and also to illustrate new ideas.
Categories: Mathematics

# Class Groups and Picard Groups of Group Rings and Orders

The aim of the lectures is to provide an introduction to recent developments in the theory of class groups and Picard groups.

Author: Irving Reiner

Publisher: American Mathematical Soc.

ISBN: 9780821816769

Category: Mathematics

Page: 44

View: 247

The aim of the lectures is to provide an introduction to recent developments in the theory of class groups and Picard groups. The techniques employed come from the three main areas: algebraic number theory, representation theory of algebras and orders, and algebraic $K$-theory.
Categories: Mathematics

# Classgroups of Group Rings

group rings. There is an excellent account of the level of knowledge obtained by
such methods in S. Ullom's survey article ... He was able to show that such
classgroups are naturally isomorphic to the quotient of two groups of
homomorphisms ...

Author: Martin J. Taylor

Publisher: Cambridge University Press

ISBN: 9780521278706

Category: Mathematics

Page: 119

View: 894

This book is a self-contained account of the theory of classgroups of group rings. The guiding philosophy has been to describe all the basic properties of such classgroups in terms of character functions. This point of view is due to A. Frohlich and it achieves a considerable simplification and clarity over previous techniques. A main feature of the book is the introduction of the author's group logarithm, with numerous examples of its application. The main results dealt with are: Ullom's conjecture for Swan modules of p-groups; the self-duality theorem for rings of integers of tame extensions; the fixed-point theorem for determinants of group rings; the existence of Adams operations on classgroups. In addition, the author includes a number of calculations of classgroups of specific families of groups such as generalized dihedral groups, and quaternion and dihedral 2-groups. The work contained in this book should be readily accessible to any graduate student in pure mathematics who has taken a course in the representation theory of finite groups. It will also be of interest to number theorists and algebraic topologists.
Categories: Mathematics

# Modules over the Integral Group Ring of a Non Abelian Group of Order pq

By using pullbacks, we obtain a description of finitely generated modules over the integral group ring of a non-abelian group of order [italic]pq.

Author: Lee Klingler

Publisher: American Mathematical Soc.

ISBN: 9780821823439

Category: Mathematics

Page: 125

View: 184

If $p,q$ are primes with $p\equiv 1$ $({\rm mod}\,q)$ there is a unique nonabelian group $G$ of order $pq$. The author obtains a classification of the finitely generated $ZG$-modules in sufficient detail to show (1) the Krull-Schmidt theorem fails but the number of indecomposable summands of a decomposable module is independent of the decomposition. He determines (2) the extent to which cancellation of direct summands $(A\oplus B\approx A\oplus C$ implying $B\approx C)$ is valid. He finds (3) the obstruction (in an appropriate semigroup) to the isomorphism of two modules which have isomorphic $t$-adic completions for each prime $t$. The indecomposable modules are constructed by expressing the ring $ZG$ as a pull-back.
Categories: Mathematics

# An Introduction to Group Rings

149] K.W. Roggenkamp and L.L. Scott, Isomorphisms for p-adic group rings, Ann.
Math. 126 (1987), 593-647. 150] K.W. Roggenkamp and M.J. Taylor, Group
Rings and Class Groups, Birkhauser, Basel, Boston and Berlin, 1992. 151] J.

Author: César Polcino Milies

Publisher: Springer Science & Business Media

ISBN: 1402002386

Category: Mathematics

Page: 371

View: 745

to Group Rings by Cesar Polcino Milies Instituto de Matematica e Estatistica, Universidade de sao Paulo, sao Paulo, Brasil and Sudarshan K. Sehgal Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton. Canada SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-0239-7 ISBN 978-94-010-0405-3 (eBook) DOI 10.1007/978-94-010-0405-3 Printed an acid-free paper AII Rights Reserved (c) 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint ofthe hardcover Ist edition 2002 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inc1uding photocopying, recording Of by any information storage and retrieval system, without written permis sion from the copyright owner. Contents Preface ix 1 Groups 1 1.1 Basic Concepts . . . . . . . . . . . . 1 1.2 Homomorphisms and Factor Groups 10 1.3 Abelian Groups . 18 1.4 Group Actions, p-groups and Sylow Subgroups 21 1.5 Solvable and Nilpotent Groups 27 1.6 FC Groups .
Categories: Mathematics

# Logarithmic Descriptions of Whitehead Groups and Class Groups for p Groups

M. Taylor, Locally free classgroups of groups of prime power order, J. Algebra 50
(1978), 463-487. M. Taylor, A logarithmic approach to classgroups of integral
group rings, J. Alg. 66 (1980), 321–353. S. Ullom, The exponent of class groups,
...

Author: Robert Oliver

Publisher: American Mathematical Soc.

ISBN: 9780821824559

Category: Mathematics

Page: 97

View: 307

P-adic logarithms are used to translate localization sequences, involving multiplicative groups of units, to simpler additive descriptions of [italic]D([double-struck]Z[italic]G) and [bold]Wh′([italic]G) for the [italic]p-group [italic]G. The goal of this paper is to work out a procedure for using logarithms to translate the localization sequences in to sequences involving additive groups.
Categories: Mathematics

# Artinian Modules over Group Rings

In this book we will consider modules over group rings when the groups have
certain features, for example, are near to ... In the theory of infinite groups, many
types of groups appear in studying cocentralizers of their conjugacy classes. This
is ...

Author: Leonid Kurdachenko

Publisher: Springer Science & Business Media

ISBN: 9783764377656

Category: Mathematics

Page: 247

View: 221

This book highlights important developments on artinian modules over group rings of generalized nilpotent groups. Along with traditional topics such as direct decompositions of artinian modules, criteria of complementability for some important modules, and criteria of semisimplicity of artinian modules, it also focuses on recent advanced results on these matters.
Categories: Mathematics

# Groups Rings and Group Rings

International Conference : Groups, Rings, and Group Rings, July 28-August 2,
2008, Ubatuba, Brazil A. Giambruno, César ... Introduction Within the last decade
there has been a growing interest in the class of finitely presented semigroup ...

Author: A. Giambruno

Publisher: American Mathematical Soc.

ISBN: 9780821847718

Category: Mathematics

Page: 270

View: 895

This volume represents the proceedings of the conference on Groups, Rings and Group Rings, held July 28 - August 2, 2008, in Ubatuba, Brazil. Papers in this volume contain results in active research areas in the theory of groups, group rings and algebras (including noncommutative rings), polynomial identities, Lie algebras and superalgebras. In particular, topics such as growth functions on varieties, groups of units in group rings, representation theory of Lie algebras, Jordan, alternative and Leibniz algebras, graded identities, automorphisms of trees, and partial actions, are discussed.
Categories: Mathematics

# Commutative Ring Theory

" Exploring commutative algebra's connections with and applications to topological algebra and algebraic geometry, Commutative Ring Theory covers the spectra of rings chain conditions, dimension theory, and Jaffard rings fiber products ...

Author: Cahen

Publisher: CRC Press

ISBN: 0824791703

Category: Mathematics

Page: 280

View: 361

" Exploring commutative algebra's connections with and applications to topological algebra and algebraic geometry, Commutative Ring Theory covers the spectra of rings chain conditions, dimension theory, and Jaffard rings fiber products group rings, semigroup rings, and graded rings class groups linear groups integer-valued polynomials rings of finite fractions big Cohen-Macaulay modules and much more!"
Categories: Mathematics

# Commutative Semigroup Rings

C. Ayoub, Restricted chain conditions on groups and rings, Houston J. Math. ...
R. G. Ayoub and C. Ayoub, On the group ring of a finite abelian group, Bull.
Austral. Math. ... L. G. Chouinard II, Krull semigroups and divisor class groups,

Author: Robert Gilmer

Publisher: University of Chicago Press

ISBN: 0226293920

Category: Mathematics

Page: 380

View: 118

Commutative Semigroup Rings was the first exposition of the basic properties of semigroup rings. Gilmer concentrates on the interplay between semigroups and rings, thereby illuminating both of these important concepts in modern algebra.
Categories: Mathematics

# Unit Groups of Group Rings

1965 Some properties of the group ring over rational integers of a finite group ,
Notices Amer Math Soc 12 , 463 . Taussky O ... Ullom S 1970 A note on the class
group of integral group rings of some cyclic groups , Mathematika 17 , 74 - 81 .

Author: Gregory Karpilovsky

Publisher: Longman Scientific and Technical

ISBN: UOM:39015015498077

Category: Commutative rings

Page: 393

View: 262

Categories: Commutative rings

# Abstract Algebra

Suitable for second to fourth year undergraduates, this title contains several applications: Polya-Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes and a classification of the finite groups of isometries of ...

Author: Clive Reis

Publisher: World Scientific

ISBN: 9789814335645

Category: Mathematics

Page: 491

View: 682

Suitable for second to fourth year undergraduates, this title contains several applications: Polya-Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space.
Categories: Mathematics

# Noncommutative Rings Group Rings Diagram Algebras and Their Applications

Articles in this volume are based on talks given at the International Conference on Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications.

Author: Dinesh Khurana

Publisher: American Mathematical Soc.

ISBN: 9780821842850

Category: Mathematics

Page: 230

View: 936

Articles in this volume are based on talks given at the International Conference on Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications. The conference provided researchers in mathematics with the opportunity to discuss new developments in these rapidly growing fields. This book contains several excellent articles, both expository and original, with new and significant results. It is suitable for graduate students and researchers interested in Ring Theory, Diagram Algebras and related topics.
Categories: Mathematics

# Groups Korea 1983

[ 2 ] A . Fröhlich , " The Picard group of non - commutative rings , in particular of
orders " , Trans . Amer . Math . Soc . 180 ( 1973 ) , pp . 1 - 45 , MR47 : 6751 . [ 3 ]
A . Fröhlich , I . Reiner , and S . Ullom , " Class groups of orders and Picard
groups " ...

Author: A.C. Kim

Publisher: Springer

ISBN: 3540138900

Category: Mathematics

Page: 190

View: 226

Categories: Mathematics

# Group and Semigroup Rings

Moreover, such group rings are shown to be strongly Lie metabelian and strongly
Lie nilpotent of class at most 3. 1. For any associative ring R we ... groups G
which have a Lie metabelian group ring RG, that is, s!?! (RG) = 0. For a general ...

Author: G. Karpilovsky

Publisher: Elsevier

ISBN: 0080872379

Category: Mathematics

Page: 266

View: 832

A broad range of topics is covered here, including commutative monoid rings, the Jacobson radical of semigroup rings, blocks of modular group algebras, nilpotency index of the radical of group algebras, the isomorphism problem for group rings, inverse semigroup algebras and the Picard group of an abelian group ring. The survey lectures provide an up-to-date account of the current state of the subject and form a comprehensive introduction for intending researchers.
Categories: Mathematics

# An Algebraic Introduction to K Theory

These are groups, rings, or modules whose elements are themselves
isomorphism classes of groups, rings, ... It focuses on a sequence of abelian
groups Kn(R) associated to each ring ft The first of these is Ko(R), Grothendieck's
group ...

Author: Bruce A. Magurn

Publisher: Cambridge University Press

ISBN: 9781107079441

Category: Mathematics

Page:

View: 816

This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs.
Categories: Mathematics

# Sylow Theory Formations and Fitting Classes in Locally Finite Groups

It also contains details of Sunkov's and Belyaev'ss results on locally finite groups with min-p for all primes p. This is the first time many of these topics have appeared in book form. The body of work here is fairly complete.

Author: Martyn Russell Dixon

Publisher: World Scientific

ISBN: 9810217951

Category: Mathematics

Page: 304

View: 176

This book is concerned with the generalizations of Sylow theorems and the related topics of formations and the fitting of classes to locally finite groups. It also contains details of Sunkov's and Belyaev'ss results on locally finite groups with min-p for all primes p. This is the first time many of these topics have appeared in book form. The body of work here is fairly complete.
Categories: Mathematics

# Introductory Lectures on Rings and Modules

The main result on finite abelian groups is the structure theorem : every finite
abelian group is isomorphic to a direct sum ... 2.1 Basic definitions and examples
Just as permutation groups form the generic class of groups , endomorphism
rings ...

Author: John A. Beachy

Publisher: Cambridge University Press

ISBN: 0521644070

Category: Mathematics

Page: 238

View: 591

Categories: Mathematics

# Some Recent Developments in Operator Theory

W. Gruenberg: Relation modules of finite groups 26. Irving Reiner: Class groups
and Picord groups of group rings and orders 27. H. Blaine Lawson, Jr.: The
quantitative theory of foliattons 28. T. A. Chapman: Lectures on Hilbert cube
manifolds ...

Author: Carl M. Pearcy

Publisher: American Mathematical Soc.

ISBN: 0821888811

Category: Mathematics

Page: 73

View: 645

Categories: Mathematics

# Galois Module Structure

In preparation for the intervention of these analytic invariants in the Hom -
description class - groups , their behaviour under the Galois action of NQ is ... In
Section 2.1 we recall the Hom - description of the class - group of a group - ring .

Author: Victor Percy Snaith

Publisher: American Mathematical Soc.

ISBN: 0821871781

Category: Mathematics

Page: 207

View: 854

This is the first published graduate course on the Chinburg conjectures, and this book provides the necessary background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions. The computation of Hom-descriptions is facilitated by Snaith's Explicit Brauer Induction technique in representation theory. In this way, illustrative special cases of the main results and new examples of the conjectures are proved and amplified by numerous exercises and research problems.
Categories: Mathematics