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.

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Correspondence to S. De Winter.

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The author is Postdoctoral Fellow of the Fund for Scientific Research Flanders (FWO-Vlaanderen).

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Winter, S.D. Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry. J Algebr Comb 24, 285–297 (2006). https://doi.org/10.1007/s10801-006-0019-2

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Keywords

  • Partial geometry
  • Abelian Singer group
  • Geometry of van Lint-Schrijver