On pairs of Baer subplanes in \(PG(2,q^2)\)
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.
scientific article; zbMATH DE number 1268133