Ein gruppentheoretischer Aufbau der äquiformen Geometrie
type: Item
instance of: scholarly article
title: Ein gruppentheoretischer Aufbau der äquiformen Geometrie
zbMATH Open document ID: 0565.51004
author: Horst Struve
published in: Beiträge zur Algebra und Geometrie
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/138303
review text: The author studies the group of similarities of a euclidean, a Galilean, and a Minkowskian plane, i.e. an affine plane over a field K of characteristic \(\neq 2\), together with an orthogonality relation defined via a symmetric bilinear form \(((x_ 1,x_ 2),(y_ 1,y_ 2))\mapsto x_ 1y_ 1+kx_ 2y_ 2\) with some \(k\in K\) (-k not a square, \(k=0\), - k a square in \(K\setminus \{0\}\), respectively, gives a euclidean, a Galilean, a Minkowskian plane, respectively). A system of seven axioms describes these groups where the distinct types of planes are distinguished by two further axioms and their negations. With these two axioms the author obtains a euclidean Bachmann group, a parabolic Hjelmslev group, or a Minkowskian group, respectively. To each of these groups well known representation theorems apply [see \textit{F. Bachmann}, ''Aufbau der Geometrie aus dem Spiegelungsbegriff'' (1973; Zbl 0254.50001), \textit{H. Struve}, ''Ein spiegelungsgeometrischer Aufbau der Galileischen Geometrie'', to appear in the same journal, and \textit{H. Wolff}, Math. Ann. 171, 144-193 (1967; Zbl 0148.146), respectively]. Since the group of similarities is not generated by reflections, some additional work is required.
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51F20
Mathematics Subject Classification ID: 20H05
zbMATH DE Number: 3901537
zbMATH Keywords: euclidean plane
zbMATH Keywords: Galilean plane
zbMATH Keywords: group of similarities
zbMATH Keywords: Minkowskian plane
zbMATH Keywords: euclidean Bachmann group
zbMATH Keywords: parabolic Hjelmslev group
zbMATH Keywords: Minkowskian group
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3901537