Finite elation Laguerre planes admitting a two-transitive group on their set of generators
type: Item
instance of: scholarly article
title: Finite elation Laguerre planes admitting a two-transitive group on their set of generators
zbMATH Open document ID: 1307.51005
author: Günter F. Steinke
author: Markus Stroppel
published in: Innovations in Incidence Geometry
publication date: 2014-06-24T00:00:00Z
review text: This paper investigates finite elation Laguerre planes admitting a group of automorphisms that is two-transitive on the set of generators. Recall that finite elation Laguerre planes were introduced in [\textit{G. F. Steinke}, Forum Math. 2, No. 3, 233--247 (1990; Zbl 0696.51008); J. Geom. 41, No. 1--2, 162--179 (1991; Zbl 0737.51003)]. They are characterized by the existence of a group of automorphisms that acts trivially on the set of generators and regularly, i.e., sharply transitively, on the set of circles. Recall also that all known finite Laguerre planes of odd order are Miquelian, i.e., arising from a cone over a conic in a projective space \(\mathrm{PG}(3,q)\). While all known finite Laguerre planes of even order are ovoidal, i.e., arising from a cone over an oval in a projective space \(\mathrm{PG}(3,q)\). Every ovoidal Laguerre plane is an elation Laguerre plane, while the vice versa is false in general, see for example G. F. Steinke [Zbl 0696.51008]. The main result of the paper states that if the automorphism group of an elation Laguerre plane of even order is doubly transitive on the set of generators, then the Laguerre plane is Miquelian. Moreover it is proved that if the automorphism group of an elation Laguerre plane of order \(q\) contains a subgroup \(\Gamma\) fixing a circle and acting two-transitively on that circle, then the Laguerre plane is Miquelian. The socle of the group induced on the fixed circle is either isomoprhic to \(\mathrm{PSL}(2,q)\) or it is \(q=4\) and the socle is isomorphic to \(\mathrm{AGL}(1,q)\). Since each translation generalized quadrangle of order \(q\) arises from a generalized oval in \(\mathrm{PG}(3n-1,r)\), \(q=r^n\), see [\textit{S. E. Payne} and \textit{J. A. Thas}, Simon Stevin 49, 3--32 (1975; Zbl 0328.50017)], the correspondence between translation generalized quadrangles of order \(q\) and elation Laguerre planes of order \(q\) immediately allows to state the following: if the automorphism group of a generalize…
reviewed by: Gloria Rinaldi
Mathematics Subject Classification ID: 51E25
Mathematics Subject Classification ID: 51B15
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51A10
zbMATH DE Number: 6308057
zbMATH Keywords: Laguerre planes
zbMATH Keywords: elation group
zbMATH Keywords: translation generalized quadrangle
zbMATH Keywords: oval
zbMATH Keywords: generalized oval
zbMATH Keywords: two-transitive group
zbMATH Keywords: socle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6308057