Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

Abstract

Let $${\mathcal{S}}$$ be a proper partial geometry pg(s,t,2), and let G be an abelian group of automorphisms of $${\mathcal{S}}$$ acting regularly on the points of $${\mathcal{S}}$$ . Then either t≡2±od s+1 or $${\mathcal{S}}$$ is a pg(5,5,2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63–73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.