Factoring Abelian Groups, Cliques in Graphs and Covering Sets (original) (raw)

Factorization of periodic subsets

Acta Mathematica Hungarica, 1991

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.

A type of factorization of finite Abelian groups

Discrete Mathematics, 1985

A clique is a subgraph in a graph that is complete in the sense that each two of its nodes are connected by an edge. Finding cliques in a given graph is an important procedure in discrete mathematical modeling. The paper will show how concepts such as splitting partitions, quasi coloring, node and edge dominance are related to clique search problems. In particular we will discuss the connection with parallel clique search algorithms. These concepts also suggest practical guide lines to inspect a given graph before starting a large scale search.

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 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.

2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs

We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B, transposed by an automorphism of order 2. The case where G is metacyclic having been dealt with elsewhere, we show that for each e ≥ 3 there are exactly three such non-metacyclic groups G with |A| = |B| = 2 e , and for e = 2 there is one. These groups appear in a classification by Berkovich and Janko of 2-groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups.

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.

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 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.