Sándor Szabó

Factoring elementary groups of prime cube order into subsets

Let p be a prime and let G be the 3-fold direct product of the cyclic group of order p. Redei conjectured if G is the direct product of subsets A and B, each of which contains the identity element of G, then either A or B does not generate all of G. The paper verifies Redeis conjecture for p ≤ 11.

Solution to a problem of A. D. sands

If a finite cyclic group is a direct product of its subsets such that the cardinality of one factor is a product of two primes and the others are of prime cardinalities, then at least one of the factors is a direct product of a subset and a proper subgroup of the group. This settles a 30 years old problem of A. D. Sands,.

Keller's conjecture for certain p-groups

In 1930 0. H. Keller [4] conjectured that if translates of a closed n-dimensional cube tile the n-space, then in this cube system there exist two cubes having a common (n - 1 )-dimensional face. In 1949 G. Hajos [3] gave the following group theoretical equivalent for this conjecture. If G is a finite additive abelian group and

Factoring abelian groups into uniquely complemented subsets

Abstract 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 Hajóss factorization theorem.

Factorization Results with Combinatorial Proofs

Abstract Two results on factorization of finite abelian groups are proved using combinatorial character free arguments. The first one is a weaker form of Rédeis theorem and presented only to motivate the method. The second one is an extension of Rédeis theorem for elementary 2-groups, which was originally proved by means of characters.

An extension for Hajós' theorem

Abstract Hajos theorem asserts that if a finite abelian group is expressed as a direct product of cyclic subsets of prime cardinality, then at least one of the factors must be a subgroup. (A cyclic subset is a ‘front end’ of a cyclic subgroup.) A.D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality a power of a prime, then at least one of the factors is a direct product of some subset and a nontrivial subgroup. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subset of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.

Multiple factorizations by cyclic subsets

In connection with multiple tilings of n-dimensional space by unit cubes sharing no complete (n − 1)-dimensional face it is proved that a finite abelian group can be multiple factorized by non-subgroup cyclic subsets if and only if it is neither cyclic nor the direct sum of cyclic groups of orders pa, p, …, p, respectively.