On iterated difference sets in groups

We extend a result of Stewart, Tijdeman and Ruzsa on iterated difierence sequences to groups. We give a complete answer for abelian groups, and apart from a constant, we give the best estimate for non-abelian ones.     

On the fixing sets of dihedral groups

The fixing number of a graph $\Gamma$ is the minimum number of labeled vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of $\Gamma$. The fixing set of a finite group $G$ is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to $G$. Previously, authors have studied the fixing sets of both abelian groups and symmetric groups. In this article, we determine the fixing set of the dihedral group.     

On affine groups admitting invariant two-point sets

A two-point set is a subset of the plane which meets every line in exactly two points. We discuss previous work on the topological symmetries of a two-point set, and show that there exist subgroups of S¹ which do not leave any two-point set invariant. Further, we show that two-point sets may be chosen to be topological groups, in which case they are also homogeneous.     

On finite products of soluble groups

Let the finite groupG =AB be the product of two soluble subgroupsA andB, and letπ be a set of primes. We investigate under which conditions for the maximal normalπ-subgroups ofA, B andG the following holds:O _π (G) ∩O _π (G) ⊆O _π (G). The second author would like to thank the Department of Mathematics of the University of Mainz and the Mathematische Forschungsinstitut Oberwolfach for their excellent hospitality during the preparation of this paper.     

On products of locally polycyclic groups

Archiv der Mathematik -     

On the maximal cardinality of half-factorial sets in cyclic groups

We consider the function μ(G), introduced by W. Narkiewicz, which associates to an abelian group G the maximal cardinality of a half-factorial subset of it. In this article, we start a systematic study of this function in the case where G is a finite cyclic group and prove several results on its behaviour. In particular, we show that the order of magnitude of this function on cyclic groups is the same as the one of the number of divisors of its cardinality. This work was supported by the Austrian Science Fund FWF (Project P16770-N12) and by the Austrian-French Program ``Amadeus 2003–2004'.     

On sets of Haar measure zero in abelian polish groups

It is shown that the concept of zero set for the Haar measure can be generalized to abelian Polish groups which are not necessarily locally compact. It turns out that these groups, in many respects, behave like locally compact groups. Suitably modified, many theorems from harmonic analysis carry over to this case. A few applications are given and some open problems are mentioned.     

On coherence of graph products of groups and Coxeter groups

Graph products of groups and Coxeter groups are defined via vertex-edge-labeled graphs. We show that if the graph has a special shape, then the corresponding group is coherent, i.e. every finitely generated subgroup is finitely presented.     

On products of graphs and regular groups

A graphX is called a graphical regular representation (GRR) of a groupG if the automorphism group ofX is regular and isomorphic toG. Watkins and Nowitz have shown that the direct productG×H of two finite groupsG andH has aGRR if both factors have aGRR and if at least one factor is different from the cyclic group of order two. We give a new proof of this result, thereby removing the restriction to finite groups. We further show that the complementX′ of a finite or infinite graphX is prime with respect to cartesian multiplication ifX is composite and not one of six exceptional graphs.     

On semidirect products and wreath products of partially ordered groups

The notions of Cartesian and semidirect products for partially ordered groups are considered. A series of results on those products of AO mathcal{A}mathcal{O} -groups and interpolation groups is obtained. Some results concerning wreath products of directed groups are obtained.