Durch Kollineationsgruppen bestimmte projektive Räume. (On projective spaces determined by collineation groups)
type: Item
instance of: scholarly article
title: Durch Kollineationsgruppen bestimmte projektive Räume. (On projective spaces determined by collineation groups)
zbMATH Open document ID: 0677.51002
published in: Beiträge zur Algebra und Geometrie
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/138421
review text: Let \({\mathcal B}\) be a basis and \({\mathcal H}^ a \)hyperplane with \({\mathcal H}\cap {\mathcal B}=\emptyset\) in the Desarguesian projective space (\({\mathcal P},{\mathcal G})\); let \(\Phi\) be the group of the projective collineations of \({\mathcal P}\) which fix the set \({\mathcal B}\) pointwise and \({\mathcal H}\) globally. If \({\mathcal M}\subset {\mathcal P}\), the incidence space (\({\mathcal M},{\mathcal G}_{{\mathcal M}})\), \({\mathcal G}_{{\mathcal M}}:=\{g\cap {\mathcal M}|\) \(g\in {\mathcal G}\), \(| g\cap {\mathcal M}| \geq 2\}\) is called a trace space of (\({\mathcal P},{\mathcal G})\). If each line of the projective space (\({\mathcal P},{\mathcal L})\) is a subset of a certain line of \({\mathcal G}_{{\mathcal M}}\), then (\({\mathcal P},{\mathcal L})\) is called a refined trace space of (\({\mathcal P},{\mathcal G})\). The author proves: Any orbit of a point in \({\mathcal P}\), under \(\Phi\), can be equipped with the structure of a projective space in such a way that it becomes a refined trace space of (\({\mathcal P},{\mathcal G})\). This ``orbit space'' may be obtained as the projection of a trace space \({\mathcal T}\) of a certain extension space of (\({\mathcal P},{\mathcal G})\), at which \({\mathcal T}\) is isomorphically mapped onto the orbit space.
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51A30
zbMATH DE Number: 4109479
zbMATH Keywords: Desarguesian projective space
zbMATH Keywords: projective collineations
author: Hans Havlicek
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4109479