On the existence of translation nets

AbstractThe existence of a translation net of order s and degree r with translation group G is equivalent to the existence of r mutually disjoint subgroups of G of order s. In this paper we consider p-groups G of odd square order p2n and improve the known general upper bound on the number of mutually disjoint subgroups of order pn in G provided that G is not elementary abelian. This solves problem 8.2.14 in (D. Jungnickel, Latin squares, their geometries and their groups. A survey, in “Coding Theory and Design Theory II” (D. K. Ray-Chaudhuri, Ed.), pp. 166–225, Springer, Berlin/New York, 1990.) We determine all groups of order p4 which are translation groups of translation nets with at least three parallel classes for all prime numbers p. Furthermore, we construct (p3, p2 + 1)-translation nets with non-abelian translation group of order p6 for all odd prime numbers p