Remarks on translation transversal designs (Q1330047)

The author studies the existence for translation transversal designs, i.e. for transversal designs (for short, TTDs) admitting a group \(G\) which acts regularly on the point set and for which the block orbits induce a parallelism. It is known that a TTD with parameters \((s,k,\lambda)\) is equivalent to a certain type of partition of \(G\) into subgroups called an \((s,k,\lambda)\)-partition. The families of groups admitting such a partition have been classified previously (by work of M. Biliotti, G. Micelli and R.-H. Schulz), but the author provides a much more elementary proof for this result which avoids the rather deep classification of the finite groups admitting a partition. He then obtains more precise information about the structure of \(p\)-groups occurring in connection with TTDs (where only a very general description was known); in particular, a complete classification is given for the case of 2-groups. Finally, an interesting series of examples of TTDs with \(\lambda= 1\) which belong to a non-Abelian \(p\)-group and admit a flag regular automorphism group is constructed.