Factorization of periodic subsets (original) (raw)

Factoring abelian groups into uniquely complemented subsets

Journal of Group Theory, 2000

The paper deals with decomposition of a finite abelian group into a direct product of subsets. A family of subsets, the so-called uniquely complemented subsets, is singled out. It will be shown that if a finite abelian group is a direct product of uniquely complemented subsets, then at least one of the factors must be a subgroup. This generalizes Hajó s's factorization theorem.

NONQUASI-PERIODIC FACTORIZATIONS FOR CERTAIN FINITE ABELIAN GROUPS

International Journal of Algebra and Computation, 2007

Hajós [2] asked if each factorization of a finite abelian group is quasi-periodic. Sands [5] exhibited a counter-example. We will show that a sizeable family of finite abelian groups admits nonquasi-periodic factorizations. We also describe a small family whose members have only quasi-periodic factorizations.

Factoring by hereditary periodicity forcing subsets

Acta Mathematica Hungarica, 2009

If a nite abelian group is factored into a direct product of its cyclic subsets, then at least one of the factors is periodic. This is a famous result of G. Hajós. We propose to replace the cyclicity of the factors by an abstract property that still guarantees that one of the factors is periodic. Then we present applications of this approach.

Factoring Finite Abelian Groups by Subsets with Maximal Span

SIAM Journal on Discrete Mathematics, 2006

The paper deals with decomposition of a finite abelian group into a direct product of subsets. A family of subsets, the so-called uniquely complemented subsets, is singled out. It will be shown that if a finite abelian group is a direct product of uniquely complemented subsets, then at least one of the factors must be a subgroup. This generalizes Hajó s's factorization theorem.

Factoring an Infinite Abelian Group by Subsets

2000

By a theorem of L. Rédei if a finite abelian group is a direct product of its subsets such that each subset has a prime number of elements and contains the identity element of the group, then at least one of the factors must be a subgroup. The content of this paper is that this result holds for certain infinite abelian groups, too. Namely, for groups that are direct products of finitely many Prüferian groups and finite cyclic groups of prime power order, belonging to pairwise distinct primes.

Complete decompositions of finite abelian groups

Applicable Algebra in Engineering, Communication and Computing, 2018

Let G be a nontrivial abelian group and let A 1 ,. .. , A h (h ≥ 2) be nonempty subsets of G. We say that A 1 ,. .. , A h is a complete decomposition of G of order h if A 1 + • • •+ A h = G and A i ∩ A j = ∅ for i, j = 1,. .. , h (i = j). In this paper we consider the case G is the cyclic group Z n and determine the values of h for which a complete decomposition of Z n of order h exists. The result is then extended to the case G is a finite abelian group. We also investigate the existence of complete decompositions of Z n where the cardinality of each set in the decomposition is a prescribed integer ≥ 2. As an application, we describe a way to construct codes over a binary alphabet using a construction of a complete decomposition of cyclic groups.

Factoring Abelian Groups, Cliques in Graphs and Covering Sets

Algebra Colloquium, 2011

The paper deals with the following problem: If a finite abelian 2-group is a direct product of its subsets of cardinality 4, does it follow that at least one of the factors is periodic? Two results are presented. In the first one, the structures of the group and the subsets are restricted but the size of the the group is not. In the second one, the group and the factors are general but the order of the group is 2 6 .