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.
University of Pecs, Pecs, Hungary University of Dundee, Dundee, Scotland, UK Rutgers University, Piscataway, New Jersey, USA
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 Redei 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