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 ?

Group Rings and Class Groups

Author: K.W. Roggenkamp

Publisher: Birkhäuser

ISBN: 303488611X

Page: 210

View: 770

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.

Class Groups and Picard Groups of Group Rings and Orders

Introduction The aim of these lectures is to provide an introduction to recent
developments in the theory of class groups and Picard groups. By way of
orientation, let us indicate where this theory fits into the general framework of
algebra.

Class Groups and Picard Groups of Group Rings and Orders

Author: Irving Reiner

Publisher: American Mathematical Soc.

ISBN: 0821816764

Page: 44

View: 953

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.

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.

An Introduction to Group Rings

Author: César Polcino Milies

Publisher: Springer Science & Business Media

ISBN: 9781402002380

Page: 371

View: 548

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 .

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,
 ...

Logarithmic Descriptions of Whitehead Groups and Class Groups for  p  Groups

Author: Robert Oliver

Publisher: American Mathematical Soc.

ISBN: 0821824554

Page: 97

View: 846

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.

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 ...

Artinian Modules over Group Rings

Author: Leonid Kurdachenko

Publisher: Springer Science & Business Media

ISBN: 3764377658

Page: 247

View: 471

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.

Representation Theory of Algebras

[ 16 ] A Sylowlike theorem for integral group rings of finite solvable groups , Arch .
Math . ... [ 28 ] K. W. Roggenkamp and M. Taylor , Group rings and class groups ,
DMV - Seminar Band 18 , Birkh " a ' auser , Basel Boston Berlin 1992 . ( 29 ) K. W.
 ...

Representation Theory of Algebras

Author: Raymundo Bautista

Publisher: American Mathematical Soc.

ISBN: 9780821803950

Page: 749

View: 439

The ICRA VII was held at Cocoyoc, Mexico, in August 1994. This was the second time that the ICRA was held in Mexico: ICRA III took place in Puebla in 1980. The 1994 conference included 62 lectures, all listed in these Proceedings. Not all contributions presented, however, appear in this book. Most papers in this volume are in final form with complete proofs, with the only exception being the paper of Leszczynski and Skowronski, Auslander algebras of tame representation type, that the editors thought useful to include.

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 ...

Groups  Rings  and Group Rings

Author: A. Giambruno

Publisher: American Mathematical Soc.

ISBN: 0821847716

Page: 270

View: 861

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.

Groups St Andrews 1981

K.R. Pearson 8 D.E. Taylor, Groups subnormal in the units of their modular group
rings, Proc. London Math. Soc. ... I. Reiner G S. Ullom, A Mayer-Victoris sequence
for class groups, J. Algebra 33 (1974), 305-342. P. Ribemboim, Rings and ...

Groups   St Andrews 1981

Author: C. M. Campbell

Publisher: Cambridge University Press

ISBN: 0521289742

Page: 360

View: 827

This book contains selected papers from the international conference 'Groups - St Andrews 1981', which was held at the University of St Andrews in July/August 1981. Its contents reflect the main topics of the conference: combinatorial group theory; infinite groups; general groups, finite or infinite; computational group theory. Four courses, each providing a five-lecture survey, given by J. Neubuser (Aachen), D. J. S. Robinson (Illinois), S. J. Tobin (Galway) and J. Wiengold (Cardiff), have been expanded into articles, forming the first part of the book. The second part consists of surveys and research articles written by other conference participants. More than two-thirds of the book is composed of survey articles providing a remarkably clear and up-to-date picture of those areas of group theory. The articles which comprise this book, together with their extensive bibliographies, will prove an invaluable tool to researchers in group theory, and, in addition, their detailed expositions make them very suitable for relevant postgraduate courses.

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 .

Unit Groups of Group Rings

Author: Gregory Karpilovsky

Publisher: Longman Scientific and Technical

ISBN:

Page: 393

View: 950

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 [U3]. ... motivated by the normal integral
basis problem, introduced a completely new description of such classgroups.

Classgroups of Group Rings

Author: Martin J. Taylor

Publisher: Cambridge University Press

ISBN: 0521278708

Page: 119

View: 198

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.

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 ...

Abstract Algebra

Author: Clive Reis

Publisher: World Scientific

ISBN: 9814335649

Page: 491

View: 283

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.

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 ...

An Algebraic Introduction to K Theory

Author: Bruce A. Magurn

Publisher: Cambridge University Press

ISBN: 1107079446

Page:

View: 583

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.

Transformation Groups

T. Petrie J. D. Randall : Transformation Groups on Manifolds , Marcell Dekker
1984 . I. Reiner S. Ullom : Remarks on class groups of integral group rings ,
Symposia Math . vol XIII , 501-516 . R. L. Rubinsztein : On the equivariant
homotopy of ...

Transformation Groups

Author: Katsuo Kawakubo

Publisher: Springer

ISBN: 9783540512189

Page: 398

View: 780

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 ...

Introductory Lectures on Rings and Modules

Author: John A. Beachy

Publisher: Cambridge University Press

ISBN: 9780521644075

Page: 238

View: 660

A first-year graduate text or reference for advanced undergraduates on noncommutative aspects of rings and modules.

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 ...

Group and Semigroup Rings

Author: G. Karpilovsky

Publisher: Elsevier

ISBN: 9780080872377

Page: 266

View: 355

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.

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 ...

Some Recent Developments in Operator Theory

Author: Carl M. Pearcy

Publisher: American Mathematical Soc.

ISBN: 9780821888810

Page: 73

View: 822

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 .

Galois Module Structure

Author: Victor Percy Snaith

Publisher: American Mathematical Soc.

ISBN: 9780821871782

Page: 207

View: 751

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.

Illinois Journal of Mathematics

H. BASS AND M.P. MURPHY , Grothendieck groups and Picard groups of
abelian group rings , Ann . of Math . , vol . ... A. FRÖHLICH , M.E. KEATING and
S.M.J. Wilson , The class group of quaternion and dihedral 2 - groups ,
Mathematica , vol ...

Illinois Journal of Mathematics

Author:

Publisher:

ISBN:

Page:

View: 604