Durch Kollineationsgruppen bestimmte projektive Räume. (On projective spaces determined by collineation groups)
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.
scientific article; zbMATH DE number 4109479