Collineations of projective planes with different numbers of fixed points and fixed lines
type: Item
instance of: scholarly article
title: Collineations of projective planes with different numbers of fixed points and fixed lines
zbMATH Open document ID: 0661.51002
DOI: 10.1016/S0195-6698(88)80068-4
author: Helmut Mäurer
published in: European Journal of Combinatorics
publication date: 1988-01-01T00:00:00Z
review text: Let \(F_ P\) and \(F_ L\) be the sets of fixed points and fixed lines of a collineation \(\kappa\) of a projective plane. For the cardinalities \(| F_ P|\) and \(| F_ L|\) it is known that \(| F_ P| =| F_ L|\) if the projective plane is pappian and \(\kappa\) is a projective collineation. The author proves that for every collineation of finite order in a desarguesian projective plane the equality \(| F_ P| =| F_ L|\) holds. There are also shown five examples in desarguesian projective planes with \(| F_ P| <| F_ L|\), having \(F_ P=\emptyset\), \(| F_ L| =l\), or \(| F_ P| =l\), \(F_ L\) the set of the lines through the fixed point, or \(| F_ P| \geq 2\), \(F_ P\) being contained in a line and \(F_ L\) a set of lines through a point of \(F_ P\). One of the examples shows that the equality \(| F_ P| =| F_ L|\) valid for a projective collineation in a pappian plane does not hold in an arbitrary desarguesian projective plane.
Mathematics Subject Classification ID: 51A30
zbMATH DE Number: 4080269
zbMATH Keywords: collineations of desarguesian projective planes
zbMATH Keywords: fixed points
zbMATH Keywords: fixed lines
MaRDI profile type: Publication
cites work: Q3241211
cites work: Q3259104
cites work: Projectivities of Finite Projective Planes
cites work: Q5817043
cites work: Q5683869
full work available at URL: s0195-6698(88)80068-4
OpenAlex ID: W1987809569
description: scientific article; zbMATH DE number 4080269