Factoring Groups into Subsets / Edition 1

by Sandor Szabo

Overview

Decomposing an abelian group into a direct sum of its subsets leads to results that can be applied to a variety of areas, such as number theory, geometry of tilings, coding theory, cryptography, graph theory, and Fourier analysis. Focusing mainly on cyclic groups, Factoring Groups into Subsets explores the factorization theory of abelian groups.

The book first shows how to construct new factorizations from old ones. The authors then discuss nonperiodic and periodic factorizations, quasiperiodicity, and the factoring of periodic subsets. They also examine how tiling plays an important role in number theory. The next several chapters cover factorizations of infinite abelian groups; combinatorics, such as Ramsey numbers, Latin squares, and complex Hadamard matrices; and connections with codes, including variable length codes, error correcting codes, and integer codes. The final chapter deals with several classical problems of Fuchs.

Encompassing many of the main areas of the factorization theory, this book explores problems in which the underlying factored group is cyclic.

Product Details

ISBN-13:	9781420090468
Publisher:	Taylor & Francis
Publication date:	01/21/2009
Series:	Lecture Notes in Pure and Applied Mathematics , #257
Pages:	286
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Sandor Szabo, Arthur D. Sands

Table of Contents

Symbol Descriptions ix

List of Tables xi

List of Figures xiii

Preface xv

1 Introduction 1

2 New factorizations from old ones 11

2.1 Restriction 11

2.2 Factorization 16

2.3 Homomorphism 22

2.4 Constructions 28

3 Non-periodic factorizations 37

3.1 Bad factorizations 37

3.2 Characters 45

3.3 Replacement 55

4 Periodic factorizations 63

4.1 Good factorizations 63

4.2 Good groups 75

4.3 Krasner factorizations 87

5 Various factorizations 93

5.1 The Rédei property 93

5.2 Quasi-periodicity 105

6 Factoring by many factors 121

6.1 Factoring periodic subsets 121

6.2 Simulated subsets 128

7 Group of integers 141

7.1 Sum sets of integers 141

7.2 Direct factor subsets 146

7.3 Tiling the integers 152

8 Infinite groups 161

8.1 Groups with cyclic subgroups 161

8.2 Groups with special p-components 169

9 Combinatorics 183

9.1 Complete maps 183

9.2 Ramsey numbers 189

9.3 Near factorizations 193

9.4 A family of random graphs 199

9.5 Complex Hadamard matrices 201

10 Codes 207

10.1 Variable length codes 207

10.2 Error correcting codes 213

10.3 Tilings 216

10.4 Integer codes 225

11 Some classical problems 235

11.1 Fuchs's problems 235

11.2 Fill-rank factorizations 239

11.3 Z-subsets 243

References 253

Index 265

What People are Saying About This

From the Publisher

The book under review was written by two leading experts in this field.… The exposition is clear and detailed—it is enriched with examples and exercises—making the book, as envisioned by the authors, readily accessible to non-experts in the field.
—Mathematical Reviews, Issue 2010h

Editorial Reviews

The book under review was written by two leading experts in this field.… The exposition is clear and detailed—it is enriched with examples and exercises—making the book, as envisioned by the authors, readily accessible to non-experts in the field.
—Mathematical Reviews, Issue 2010h

From the Publisher