On elation Laguerre planes with a two-transitive orbit on the set of generators
type: Item
instance of: scholarly article
title: On elation Laguerre planes with a two-transitive orbit on the set of generators
zbMATH Open document ID: 1396.51002
author: Günter F. Steinke
author: Markus Stroppel
published in: Finite Fields and their Applications
publication date: 2018-08-08T00:00:00Z
full work available at URL: http://hdl.handle.net/10092/15256
review text: Finite elation Laguerre planes were introduced in [the first author, J. Geom. 41, No 1--2, 162--179 (1991; Zbl. 0737.51003)]. They are characterized by the existence of a group of automorphisms acting trivially on the set of generators and sharply transitively on the set of circles. The main result of the paper is a characterization of miquelian Laguerre planes over \(\mathrm{GF}(q)\) in the case \(q\) odd. It is proved that the finite elation Laguerre planes of odd order \(q\) are precisely the miquelian Laguerre planes over \(\mathrm{GF}(q)\) when they have an automorphism group fixing a generator and acting 2-transitively on the set of remaining generators. For \(q\) even non-miquelian examples exist, but for \(q=2^e\) with \(e\) a prime or a square of a prime, the Laguerre plane is always ovoidal over a translation oval.
Mathematics Subject Classification ID: 51B15
Mathematics Subject Classification ID: 51E25
Mathematics Subject Classification ID: 20B20
zbMATH DE Number: 6914843
zbMATH Keywords: Laguerre plane
zbMATH Keywords: elation group
zbMATH Keywords: two-transitive group
zbMATH Keywords: socle
zbMATH Keywords: ovoidal Laguerre plane
zbMATH Keywords: Miquelian Laguerre plane
reviewed by: Dirk Keppens
MaRDI profile type: Publication
OpenAlex ID: W2806263702
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Wikidata QID: Q129710693
DOI: 10.1016/J.FFA.2018.05.005
description: scientific article; zbMATH DE number 6914843