Remarks on Translation Transversal Designs

Abstract

In this paper the existence of translation transversal designs which is equivalent to the existence of certain partitions in finite groups is studied. All considerations are based on the fact that the particular component of such a partition (the component representing the point classes of the corresponding design) is a normal subgroup of the translation group. With regard to groups admitting an ( s, k, λ)-partition, on one hand the already known families of such groups are determined without using R. Baer′s, O. H. Kegel′s, and M. Suzuki′s classification of finite groups with partition and on the other hand some new results on the structure of p-groups admitting an ( s, k, λ)-partition are proved. Furthermore, the existence of a series of nonabelian p-groups of odd order which can be represented as translation groups of certain ( s, k, 1)-translation transversal designs is shown; moreover, the translation groups are normal subgroups of collineation groups acting regularly on the set of flags of the same designs.