Reflection groups and \(K\)-loops
type: Item
instance of: scholarly article
title: Reflection groups and \(K\)-loops
zbMATH Open document ID: 0821.51011
DOI: 10.1007/BF01406832
author: Helmut Karzel
author: Angelika Konrad
published in: Journal of Geometry
publication date: 1995-09-19T00:00:00Z
review text: There is a vast literature on reflection groups -- that is, groups generated (or containing a subgroup generated) by reflections in lines -- as a language in which to axiomatize 2-dimensional geometry [\textit{F. Bachmann}, `Aufbau der Geometrie aus dem Spiegelungsbegriff', 2. Auflage (1973; Zbl 0254.50001) and `Ebene Spiegelungsgeometrie' (1989; Zbl 0681.51001), \textit{R. Lingenberg}, `Metric planes and metric vector spaces' (1979; Zbl 0419.51001)]. By contrast, very little has been written on geometries determined by groups generated by reflections in points [\textit{K. Prazmowski}, Mitt. Math. Semin. Giessen 193, 107 pp. (1989; Zbl 0684.51002), \textit{C. F. Manara} and \textit{M. Marchi}, Rend. Sci. Mat. Appl. A 125, No. 2, 203-217 (1991; Zbl 0798.51017)], \textit{H. Hotje, M. Marchi} and \textit{S. Pianta}, Discrete Math. 129, No. 1-3, 139-147 (1994; Zbl 0804.51018)], and it refers to the (generalized) Euclidean case only. The paper under review studies the geometry that can be associated to reflection groups \(\Gamma\) containing a subset \({\mathcal D}\) of point- reflections, that satisfy the axioms (1) ``The product of the reflections in three collinear points is a reflection in a point'' and (2) ``Every pair of points has a unique midpoint'' (the `line' determined by the `points' (`point' = point-reflection in that point) \(a\) and \(b\) is defined as the set of all points \(x \in {\mathcal D}\) such that \(xab\) is involutory; let \(\mathfrak G\) be the set of all lines in \(\mathcal D\); a `midpoint' of \(a\), \(b\) is a point \(m\) such that \(b = mam\)). It is shown that, by fixing a point \(o \in {\mathcal D}\), one can define an operation \(+ : {\mathcal D} \times {\mathcal D} \rightarrow {\mathcal D}\) by \(a + b := a'oboa'\), where \(a' \in {\mathcal D}\) is the midpoint of \(o\) and \(a\) (i.e. \(a = a'oa'\)), which turns \(({\mathcal D},+)\) into a \(K\)-loop (a structure introduced by the first author [Abh. Math. Semin. Univ. Hamburg 32, 191- 2…
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 51F10
Mathematics Subject Classification ID: 51F05
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 739184
zbMATH Keywords: reflection groups
zbMATH Keywords: point-reflections
zbMATH Keywords: \(K\)-loop
zbMATH Keywords: absolute space
reviewed by: Victor V. Pambuccian
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 739184