A note on Minkowski planes admitting groups of automorphisms of some large types with respect to homotheties
type: Item
instance of: scholarly article
title: A note on Minkowski planes admitting groups of automorphisms of some large types with respect to homotheties
zbMATH Open document ID: 1278.51004
author: Günter F. Steinke
published in: Journal of Geometry
publication date: 2013-04-08T00:00:00Z
review text: A Minkowski plane \({\mathcal M}= ({\mathcal P},{\mathcal C},{\mathcal G}_1,{\mathcal G}_2)\) is an incidence structure consisting of a point set \({\mathcal P}\), a circle set \({\mathcal C}\) (the elements of which are non empty subsets of \({\mathcal P}\)) and two different partitions \({\mathcal G}_1\) and \({\mathcal G}_2\) of \({\mathcal P}\) (parallel classes), whose elements are called generators, satisfying a set of defining axioms. Namely: three mutually non-parallel points, i.e., not belonging to the same generator, are joined by a unique circle; each generator meets each circle in a unique point; each generator in \({\mathcal G}_1\) meets each generator in \({\mathcal G}_2\) in exactly one point; there is a circle that contains at least three points; the circles which touch a fixed circle \(K\) at a point \(p\in K\) partition \({\mathcal P}- ([p]_1\cup [p]_2)\). Here \([p]_i\) denotes the parallel class in \({\mathcal G}_i\) that contains \(p\). The incidence structure whose point set is \({\mathcal P}\) minus all points of \([p]_1\) and \([p]_2\), and whose lines are obtained by all circles through \(p\) and by all the generators, except \([p]_1\) and \([p]_2,\) is an affine plane, the so called derived plane at \(p\). A Minkwsi plane is of finite order when it has a finite number of points. Examples of Minkowski planes are obtained over a field \(\mathbb F\) as the geometry of non-trivial plane sections of a ruled quadric in \(3\)-dimensional projective space over \(\mathbb F\). Such planes are referred to as Miquelian planes. An automorphism of a Minkowski plane is a permutation of the point-set mapping circles onto circles and generators onto generators. An automorphism \(\gamma\) is a \(\{p,q\}\)-homothety if \(\gamma\) fixes \(p\) and \(q\) and induces a homothety with centre \(q\) in the derived plane at \(p\). In [\textit{M. Klein}, J. Geom. 43, No. 1--2, 116--128 (1992; Zbl 0746.51009)], a classification of Minkowski planes with respect to linea…
reviewed by: Gloria Rinaldi
Mathematics Subject Classification ID: 51B20
Mathematics Subject Classification ID: 51E25
zbMATH DE Number: 6151649
zbMATH Keywords: Minkowski plane
zbMATH Keywords: Minkowski homothety
zbMATH Keywords: linear transitive group of central automorphisms
zbMATH Keywords: Klein type
MaRDI profile type: Publication
full work available at URL: s00022-013-0143-9
OpenAlex ID: W2006058387
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DOI: 10.1007/S00022-013-0143-9
description: scientific article; zbMATH DE number 6151649