Finite elation Laguerre planes admitting a two-transitive group on their set of generators

Abstract

We investigate finite elation Laguerre planes admitting a group of automorphisms that is two-transitive on the set of generators. We exclude the sporadic cases of socles in two-transitive groups, as well as the alternating and Suzuki groups and the cases with abelian socle (except for the smallest ones, where the Laguerre planes are Miquelian of order at most four). The remaining cases are dealt with in a separate paper. We prove that a finite elation Laguerre plane is Miquelian if its automorphism group is two-transitive on the set of generators. Equivalently, each translation generalized quadrangle of order $q$ with a group of automorphisms acting two-transitively on the set of lines through the base point is classical.