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El. knyga: Factoring Groups into Subsets

(University of Pecs, Pecs, Hungary)
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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.

Recenzijos

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

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 Redei 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 Full-rank factorizations
239
11.3 Z-subsets
243
References 253
Index 265
University of Pecs, Pecs, Hungary University of Dundee, Dundee, Scotland, UK Rutgers University, Piscataway, New Jersey, USA
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