Factoring Groups into Subsets

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Features

  • Discusses the classification of cyclic groups with periodic factorizations and non-full-rank factorizations
  • Covers quasiperiodicity and the factoring of the group of integers
  • Provides a self-contained treatment of the more general theory of factorization so practitioners do not have to immerse themselves in the details of the full generality
  • Presents applications to variable length codes and integer codes
  • Includes many exercises throughout

Summary

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.

Table of Contents

Introduction

New Factorizations from Old Ones

Restriction

Factorization

Homomorphism

Constructions

Nonperiodic Factorizations

Bad factorizations

Characters

Replacement

Periodic Factorizations

Good factorizations

Good groups

Krasner factorizations

Various Factorizations

The Rédei property

Quasiperiodicity

Factoring by Many Factors

Factoring periodic subsets

Simulated subsets

Group of Integers

Sum sets of integers

Direct factor subsets

Tiling the integers

Infinite Groups

Cyclic subgroups

Special p-components

Combinatorics

Complete maps

Ramsey numbers

Near factorizations

A family of random graphs

Complex Hadamard matrices

Codes

Variable length codes

Error correcting codes

Tilings

Integer codes

Some Classical Problems

Fuchs’s problems

Full-rank factorizations

Z-subsets

References

Index

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