Finite elation Laguerre planes admitting a two-transitive group on their set of generators
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 generalized quadrangle of order \(q\) is two-transitive on the set of lines through the base point, then the quadrangle is isomorphic to the classical generalized quadrangle \(Q(4,q)\).
scientific article; zbMATH DE number 6308057