Constructions of large translation nets with nonabelian translation groups

In this paper the first infinite series of translation nets with nonabelian translation groups and a large number of parallel classes are constructed. For that purpose we investigate partial congruence partitions (PCPs) with at least one normal component. Two series correspond to partial congruence partitions containing one normal elementary abelian component. The construction results by using some basic facts about the first cohomology group of the translation group G regarded as an extension of the normal component which itself is a group of central translations. The other series correspond to partial congruence partitions containing two normal nonabelian components. The constructions are based on the well known automorphism method which leads to so-called splitting translation nets. By investigating the Suzuki groups Sz(q), the projective unitary groups PSU(3, q^2) and the Ree groups R(q) as doubly transitive permutation groups, we obtain examples of nonabelian groups admitting a large number of pairwise orthogonal fixed-point-free group automorphisms