Characterizations of Miquelian Laguerre planes
type: Item
instance of: scholarly article
title: Characterizations of Miquelian Laguerre planes
zbMATH Open document ID: 0799.51004
DOI: 10.1007/BF01263537
author: Paolo Gronchi
published in: Geometriae Dedicata
publication date: 1994-11-09T00:00:00Z
review text: The Lenz-Barlotti classes of projective planes classify these planes according to the configuration of point-line pairs \((p,L)\) for which the group of central collineation \(C(p,L)\) with centre \(p\) and axis \(L\) is linearly transitive, that is, transitive on \(M \backslash \{p,M \cap L\}\) for each line \(M \neq L\) through \(p\). In the Desarguesian projective planes \(C(p,L)\) is linearly transitive for every possible point-line pair \((p,L)\). However, the linear transitivity of \(C(p,L)\) for much fewer point-line pairs suffices in order to characterize the Desarguesian planes. In a similar fashion \textit{R. Kleinewillinghöfer} [`Eine Klassifikation der Laguerre-Ebenen', Diss. Darmstadt (1979; Zbl 0436.51007) and Arch. Math. 34, 469-480 (1980; Zbl 0457.51010)] obtained a classification of Laguerre planes with respect to central automorphisms and \textit{E. Hartmann} [J. Geom. 18, 9-27 (1982; Zbl 0504.51008)] gave, among other results, characterizations of Miquelian Laguerre planes by the circular transitivity of certain groups of central automorphisms of the Laguerre plane. In the paper under review the author deals with two such characterizations, one improving on one given by \textit{E. Hartmann} [loc. cit.], the other answering a conjecture in the same paper in the affirmative. The first result deals with \(L\)-translations of a Laguerre plane \(L\), that is, automorphisms of \(L\) that fix one parallel class \(P\) pointwise and one tangent pencil of circles through \(P\) elementwise. It is shown that a Laguerre plane \(L\) is Miquelian of characteristic \(\neq 2\) if and only if each group of \(L\)-translations is circularly transitive (that is, the \(L\) is of Kleinewillinghöfer type at least \(K)\) and at least one of these groups does not contain an involution. Secondly, \(L\)- dilatations are considered; these are automorphisms of \(L\) that fix a point \(P\) and a circle \(z\) through \(P\) and induce an homology with axis a parallel class \(e\) …
Mathematics Subject Classification ID: 51B15
zbMATH DE Number: 553941
zbMATH Keywords: central automorphism
zbMATH Keywords: circular transitive group
zbMATH Keywords: Miquelian Laguerre planes
zbMATH Keywords: characterizations
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
cites work: Q5674022
cites work: Über Moufang-Ovale
cites work: Transitivitaetsaetze für Laguerre-Ebenen
cites work: Q4064611
full work available at URL: https://doi.org/10.1007/bf01263537
OpenAlex ID: W1996789539
description: scientific article; zbMATH DE number 553941