A geometric characterization of elation Laguerre planes
type: Item
instance of: scholarly article
title: A geometric characterization of elation Laguerre planes
zbMATH Open document ID: 1004.51005
author: Norbert Knarr
published in: Archiv der Mathematik
publication date: 2002-05-21T00:00:00Z
review text: An elation Laguerre plane is a Laguerre that admits a group of automorphisms in the kernel of the plane that acts regularly on the set of circles. The author shows that it suffices to require that for any two nonparallel points \(p\) and \(q\) the subgroup \(\Sigma_{[p,q]}\) in the kernel that fixes each point on the parallel classes of \(p\) and \(q\) acts regularly on the set of circles through \(p\) and \(q\), or equivalently, on each parallel class containing neither \(p\) nor \(q\). Generalising a result of \textit{A. E. Schroth} [Abh. Math. Semin. Univ. Hamb. 69, 211-227 (1999; Zbl 0953.51003)] the author further characterises elation Laguerre planes as the weakly Miquelian Laguerre planes, that is, those Laguerre planes in which a certain variation M2 of Miquel's configuration involving four pairs of parallel points is satisfied.
Mathematics Subject Classification ID: 51B15
zbMATH DE Number: 1742832
zbMATH Keywords: Laguerre plane
zbMATH Keywords: elation group
zbMATH Keywords: weakly Miquelian plane
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
full work available at URL: s00013-002-8230-4
OpenAlex ID: W2043820419
DOI: 10.1007/S00013-002-8230-4
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description: scientific article; zbMATH DE number 1742832