On pairs of Baer subplanes in \(PG(2,q^2)\)
type: Item
instance of: scholarly article
title: On pairs of Baer subplanes in \(PG(2,q^2)\)
zbMATH Open document ID: 0927.51002
DOI: 10.1007/BF01221238
author: J. Eisfeld
published in: Journal of Geometry
publication date: 1999-12-13T00:00:00Z
review text: A subplane of order \(q\) in \(\Gamma G(2,q^2)\) is a Baer subplane. The author considers orbits of points of Baer subplanes under the full collineation group. \(PG(2,q^2)\) admits a ``linear partition'' into Baer subplanes: each point belongs to exactly one component of the partition. The group \(PGL(3,q^2)\) acts transitively on pairs \(({\mathcal P},M)\) where \({\mathcal P}\) is a Baer subplane and \(M\) is a linear Baer partition including \({\mathcal P}\). Two points \(({\mathcal P}_1,{\mathcal P}_2)\) and \(({\mathcal P}_3,{\mathcal P}_4)\) are said to be equivalent if there is a collineation \(\phi\) such that \({\mathcal P}\cdot{\mathcal P}_1\phi= {\mathcal P}_3\) and \({\mathcal P}_2\phi= {\mathcal P}_4\). If \({\mathcal P}_1\) and \({\mathcal P}_2\) are Baer subplanes of \(PG(2,q^2)\) we denote by \(\Gamma({\mathcal P}_1,{\mathcal P}_2)\) the incidence structure whose points and lines are the points and lines of \({\mathcal P}_1\cup{\mathcal P}_2\) and incidence is inherited from \(PG(2,q^2)\). Two points \(({\mathcal P}_1,{\mathcal P}_2)\) and \(({\mathcal P}_3,{\mathcal P}_4)\) are weakly incidence equivalent if there is an isomorphism from \(\Gamma({\mathcal P}_1,{\mathcal P}_2)\) onto \(\Gamma({\mathcal P}_3,{\mathcal P}_4)\) which moves \({\mathcal P}_1\) correspond to \({\mathcal P}_3\) and \({\mathcal P}_4\) correspond to \({\mathcal P}_2\). The author proves that every two points are weakly incidence equivalent.
reviewed by: Theodore G. Ostrom
Mathematics Subject Classification ID: 51A30
zbMATH DE Number: 1268133
zbMATH Keywords: Baer subplane
zbMATH Keywords: weakly incidence equivalent
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01221238
OpenAlex ID: W2141758041
description: scientific article; zbMATH DE number 1268133