Factoring Groups into Subsets

Sandor Szabo, Arthur D. Sands

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January 21, 2009 by Chapman and Hall/CRC
Reference - 274 Pages - 14 B/W Illustrations
ISBN 9781420090468 - CAT# C9046
Series: Lecture Notes in Pure and Applied Mathematics

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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.