Existence results for translation nets. II (Q1119651)

Let D be a translation (Bruck) net of degree s and order r (i.e., D admits a point regular group respecting parallel classes). If \(s=q_ 1...q_ n\) is the prime power factorization of s, then \(r\leq \min \{q_ i+1:\) \(i=1,...,n\}\); moreover, equality can be realized (e.g. for direct products of elementary abelian groups). Similar results are obtained for translation (s,r,\(\mu)\)-nets and, more generally, for groups with large systems of large subgroups. As a corollary, we considerably strengthen a result on symmetric translation nets by showing that every (s,r,\(\mu)\)- translation net with \(r\geq s\mu\) has parameters of the form \(s=p^ i\) and \(\mu =p^ j\) for some prime p and also necessarily has an elementary abelian translation group. These results settle some problems and conjectures from part I of the paper [Finite geometries and designs, Proc. 2nd Isle of Thorns Conf. 1980, Lond. Math. Soc., Lect. Note Ser. 49, 172-196 (1981; Zbl 0457.05011)].