Benz planes of Dembowski type and groups of tangential projectivities
type: Item
instance of: scholarly article
title: Benz planes of Dembowski type and groups of tangential projectivities
zbMATH Open document ID: 1259.51003
author: A. Barlotti
author: Karl Strambach
published in: Journal of Geometry
publication date: 2012-04-12T00:00:00Z
review text: A Benz plane \(M\) is said to be of Dembowski type if the tangency relation of \(M\) behaves like the tangency relation in a finite Benz plane of even order. For a Minkowski, respectively Möbius plane, this means that for each bundle \(\mathcal B\) of circles there exists a bundle, respectively a flock \(\mathcal B'\), such that each circle of \(\mathcal B\) is tangent to each circle of \(\mathcal B'\). For a Laguerre plane this means that for every tangent pencil \(\mathcal B\) and every point \(r\) different from the carrier of \(\mathcal B\) there exists a tangent pencil \(\mathcal B'\) with carrier \(r\) such that each circle of \(\mathcal B'\) is tangent to each circle of \(\mathcal B\). Using the tangency relation, in Benz planes of Dembowski type one can define tangential perspectivities and projectivities. The authors show that a Benz plane \(M\) of Dembowski type is Miquelian if and only if every tangential perspectivity is induced by an automorphism of \(M\). There are also several results on the group of tangential projectivities in non-Miquelian Benz planes of Dembowski type, especially in the finite case.
reviewed by: Norbert Knarr
Mathematics Subject Classification ID: 51B05
Mathematics Subject Classification ID: 51E25
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 6023282
zbMATH Keywords: Benz planes of Dembowski type
zbMATH Keywords: tangential projectivity
MaRDI profile type: Publication
full work available at URL: s00022-011-0079-x
OpenAlex ID: W2169199472
cites work: Q3241211
cites work: Collineation groups of ovals and of ovoidal Laguerre planes
cites work: Über Möbiusebenen
cites work: Das von Staudtsche Theorem in der Laguerregeometrie.
cites work: Über die Grundlagen der Laguerre-Geometrie. Ein Bericht. (On the foundations of Laguerre geometry. A report)
cites work: Kreise, Zykel, Ketten. Zur Geometrie der Algebren
cites work: Q3968056
cites work: Transfinite methods in geometry
cites work: Ovoids of PG(3,q ), q Even, with a Conic Section
cites work: Ensembles quadratiques des espaces projectifs
cites work: The Steiner system \(S(3,6,26)\)
cites work: Q5545191
cites work: The geometry of subgroups of \(PSp_4(2^n)\)
cites work: Maximal subgroups of \(PSp_4(2^n)\) containing central elations or noncentered skew elations
cites work: Schließungssätze und Projektivitäten in der Möbius- und Laguerregeometrie
cites work: Finite Fano Planes
cites work: Characterizations of Miquelian Laguerre planes
cites work: Projektivitaetengruppen von ovoidalen Moebius- und Laguerreebenen
cites work: Q3785502
cites work: Endliche Gruppen I
cites work: On a class of sharply 3-transitive groups
cites work: Classification of Minkowski planes by transitive groups of homotheties
cites work: Eine Klassifikation der Laguerre-Ebenen nach L-Streckungen und L- Translationen
cites work: Die Gruppe der eigentlichen Projektivitäten in Benz-Ebenen
cites work: Q4155413
cites work: A note on the group of projectivities of finite projective planes.
cites work: A remark on the introduction of coordinates in Minkowski planes
cites work: A geometrical picture book
cites work: Q4084884
cites work: Q3272923
cites work: On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two
cites work: Reichhaltige Untergruppen geometrischer Gruppen
cites work: Q3822294
cites work: Q4153273
DOI: 10.1007/S00022-011-0079-X
description: scientific article; zbMATH DE number 6023282