Partially sharp subsets of PΓL(n, q)

Summary

Abstract

In this article, certain translation nets which are unions of subplane covered nets are shown to be equivalent to partially sharp subsets of PFZ(n, q). In particular, translation planes of order q² and arbitrary kernel that admit two distinct Baer groups of order q–1 with identical component orbits are shown to be equivalent to partially sharp subsets of cardinality q of PTL(2, K) for some field K isomorphic to GF(q).

Introduction

Recently, a number of various connections have been established between translation planes and other geometric or combinatorial incidence structures. In particular, there are connections between flocks of hyperbolic quadrics in PG(3, q) and translation planes with spreads in PG(3, q) such that the spread is a union of reguli that share two lines. Similarly, partial flocks correspond to translation nets whose partial spreads are unions of reguli that share two lines.

It is well known that a Miquelian Minkowski plane can be defined using a hyperbolic quadric in PG(3, q) and when this connection is made, the points of the Minkowski plane are identified with the elements of PG(1, q) × PG(1, q), the circles (conies) correspond to the elements of PGL(2, q) and a flock corresponds to a sharply transitive subset of PGL(2, q).