Point-regular geometries
type: Item
instance of: scholarly article
title: Point-regular geometries
zbMATH Open document ID: 0884.51002
DOI: 10.1007/BF01229574
published in: Journal of Geometry
publication date: 1998-03-19T00:00:00Z
review text: The author continues his investigation in Geom. Dedicata 46, No. 1, 47-60 (1993; Zbl 0783.51002) of how geometries can be reconstructed from their automorphism groups. In the paper under review he considers incidence structures \((P,{\mathcal L})\) that admit sharply point transitive groups \(G\) of automorphisms. In this case, \(P\) is identified with \(G\) and \(G\) acts on itself by the right regular action. For a set \({\mathbf S}\) of subgroups of \(G\) two systems of subsets of \(G\) are defined as follows. \({\mathcal L}^{\mathbf S}=\bigcup_{H\in{\mathbf S}}G/H\) is the union of all left coset spaces \(G/H\) for \(H\) in \({\mathbf S}\). Taking right cosets \(H\setminus G\) instead he likewise defines \(^{\mathbf S} {\mathcal L}=\cup_{H\in{\mathbf S}}H\setminus G\). In the latter case, \({\mathbf S}\) consists of the stabilizers of representatives for the \(G\)-orbits on \(^{\mathbf S}{\mathcal L}\) and the whole system of lines can be thought of as moving around some standard lines; this generalizes the familiar construction of translation planes. Various properties of these two systems are investigated in the first part of the paper. Whereas the group \(G\) acts on \(^{\mathbf S} {\mathcal L}\) in a natural way, this happens to be so for \({\mathcal L}^{\mathbf S}\) if and only if \({\mathbf S}\) is invariant under conjugation (and then \(^{\mathbf S} {\mathcal L}={\mathcal L}^{\mathbf S}\)). One obtains a linear space in both situations if and only if \({\mathbf S}\) forms a group partition of \(G\), that is, every non-identity element of \(G\) is in exactly one subgroup contained in \({\mathbf S}\). Furthermore, \((G,^{\mathbf S}{\mathcal L})\) always is a sketched geometry. In fact, it is shown that every point-regular sketched geometry can be represented in this form for some collection \({\mathbf S}\) of subgroups. In the second part the author applies his method to topological planes where the point set \(P\) is a topological manifold. In this case, t…
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51A15
Mathematics Subject Classification ID: 51J99
zbMATH DE Number: 1053898
zbMATH Keywords: incidence group
zbMATH Keywords: arc plane
zbMATH Keywords: group partition
author: Markus Stroppel
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1053898