Keresztély Corrádi

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,.

Steps towards an elementary proof of frobenius' theorem

So far there has been elementary proof for Frobeniuss theorem only in special cases: if the complement is solvable, see e.g. [3], if the complement is of even order, see e.g. [6]. In the first section we consider the case, when the order of the complement is odd. We define a graph the vertices of which are the set K# of elements of our Frobenius group with 0 fixed points. Two vertices are connected with an edge if and only if the corresponding elements commute. We prove with elementary methods that K is a normal subgroup in G if and only if there exists an element x in K# such that all elements of K# belonging to the connected component C of K# containing x are at most distance 2 from c and NG(C) is not a -group, where is the set of prime divisors of the Frobenius complement of G. In the second section we generalize the case when the order of the complement is even, proving that the Frobenius kernel is a normal subgroup, if a fixed element a of the complement, the order of which is a minimal prime diviso...

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 by simulated subsets II

In earlier papers it has been shown that certain different types of conditions on the factors in a factorization of a finite abelian group by its subsets lead to the conclusion that one factor must be a subgroup. In this paper the common generalization is proved that this result still holds even if different factors satisfy different types of condition. It is also shown that one condition may be weakened without effecting the conclusion.

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.