# Factoring groups into subsets by Sandor Szabo, Arthur D. Sands By Sandor Szabo, Arthur D. Sands

Decomposing an abelian workforce right into a direct sum of its subsets ends up in effects that may be utilized to numerous components, comparable to quantity conception, geometry of tilings, coding idea, cryptography, graph conception, and Fourier research. Focusing as a rule on cyclic teams, Factoring teams into Subsets explores the factorization concept of abelian teams.

The ebook first indicates easy methods to build new factorizations from previous ones. The authors then talk about nonperiodic and periodic factorizations, quasiperiodicity, and the factoring of periodic subsets. additionally they learn how tiling performs an immense position in quantity thought. the following a number of chapters hide factorizations of limitless abelian teams; combinatorics, comparable to Ramsey numbers, Latin squares, and intricate Hadamard matrices; and connections with codes, together with variable size codes, blunders correcting codes, and integer codes. the ultimate bankruptcy bargains with numerous classical difficulties of Fuchs.

Encompassing the various major components of the factorization idea, this publication explores difficulties within which the underlying factored staff is cyclic.

Similar elementary books

How round is your circle

How do you draw a directly line? How do you identify if a circle is admittedly around? those may possibly sound like easy or perhaps trivial mathematical difficulties, yet to an engineer the solutions can suggest the variation among good fortune and failure. How around Is Your Circle? invitations readers to discover a few of the similar primary questions that operating engineers take care of each day--it's hard, hands-on, and enjoyable.

Lie Algebras and Applications

This ebook, designed for complicated graduate scholars and post-graduate researchers, introduces Lie algebras and a few in their functions to the spectroscopy of molecules, atoms, nuclei and hadrons. The publication includes many examples that aid to explain the summary algebraic definitions. It offers a precis of many formulation of functional curiosity, corresponding to the eigenvalues of Casimir operators and the size of the representations of all classical Lie algebras.

Modern Geometries

This accomplished, best-selling textual content specializes in the research of many various geometries -- instead of a unmarried geometry -- and is carefully glossy in its technique. every one bankruptcy is largely a brief path on one point of recent geometry, together with finite geometries, the geometry of variations, convexity, complex Euclidian geometry, inversion, projective geometry, geometric features of topology, and non-Euclidean geometries.

Additional info for Factoring groups into subsets

Example text

Then G = H + A + B. Now A ⊆ K and the sum K + B is direct. 1 (on page 38), it follows that A + B is non-periodic, as required. We are able to extend a bad factorization to a larger group from a subgroup. 3 If a proper subgroup H of a group G is k-bad, then G is both k-bad and (k + 1)-bad. PROOF There exists a factorization H = A1 +· · ·+Ak , where each subset Ai is non-periodic. 1 (on page 38), there exists a non-periodic subset C such that G = H + C. Then G = A1 + · · · + Ak + C and so G is © 2009 by Taylor & Francis Group, LLC Non-periodic factorizations 41 (k + 1)-bad.

Then the sum (i) A1 + B is a (|B| − 1)-fold, (ii) A + B1 is a (|A| − 1)-fold, (iii) A1 + B1 is a (|A| − 1)(|B| − 1)-fold © 2009 by Taylor & Francis Group, LLC New factorizations from old ones 25 factorization of G. PROOF (i) Note that A1 + B = (G\ A)+ B = (G+ B)\ (A+ B). The sum G + B is a |B|-fold factorization of G. The sum A + B is a 1-fold factorization of G. Therefore the sum A1 + B is a (|B| − 1)-fold factorization of G. The (ii) and (iii) cases can be settled in a similar way. We extend the concept of factorization to the case of more than two factors.

Let G be a finite abelian group and let A, B be subsets of G. If each element g ∈ G can be represented in exactly k ways in the form g = a + b, a ∈ A, b ∈ B, then we say that the sum A + B is a k-fold factorization of G. In other words, for each g ∈ G, there are k distinct pairs (a1 , b1 ), . . , (ak , bk ), ai ∈ A, bi ∈ B such that g = a1 + b1 = · · · = ak + bk . Further, g = a + b, a ∈ A, implies that (a, b) ∈ {(a1 , b1 ), . . , (ak , bk )}. 14 Let f : G → H be a homomorphism from G onto H such that k = |Kerf | is finite. 