An orbit space representation of a geometry

Abstract

When a group of automorphisms acts on a given group, a geometric structure is induced on the set of orbits. In particular, the projective space of a vector space can be viewed as a geometry of orbits. Such a geometry of orbits can be studied abstractly as a hypergroup, a multigroup or a Pasch geometry. Our approach will be the latter. A classical theorem in projective geometry states that certain abstract projective spaces are represented as orbits of vector spaces. The purpose of this paper is to generalize the socalled fundamental theorem of projective geometry and show that certain (what we have called elementary Abelian) Pasch geometries can be represented as orbits of Abelian groups. The concept of a Pasch geometry can be found in [ 3 ] or [ 2 1. For convinience we give a brief discussion here.