1st Edition

# Factoring Groups into Subsets

By Sandor Szabo, Arthur D. Sands Copyright 2009
286 Pages 14 B/W Illustrations
by Chapman & Hall

286 Pages
by Chapman & Hall

274 Pages
by Chapman & Hall

Also available as eBook on:

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.

Introduction

New Factorizations from Old Ones

Restriction

Factorization

Homomorphism

Constructions

Nonperiodic 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

Codes

Variable length codes

Error correcting codes

Tilings

Integer codes

Some Classical Problems

Fuchs’s problems

Full-rank factorizations

Z-subsets

References

Index

### Biography

Sandor Szabo, Arthur D. Sands

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