On the existence of translation nets

Abstract

The 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 p 2 n and improve the known general upper bound on the number of mutually disjoint subgroups of order p n 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 p 4 which are translation groups of translation nets with at least three parallel classes for all prime numbers p. Furthermore, we construct ( p 3, p 2 + 1)-translation nets with non-abelian translation group of order p 6 for all odd prime numbers p.