On finite projective planes in Lenz-Barlotti class at least I.3

We establish the connections between finite projective planes admitting a collineation group of Lenz–Barlotti type I.3 or I.4, partially transitive planes of type (3) in the sense of Hughes, and planes admitting a quasiregular collineation group of type (g) in the Dembowski– Piper classification; our main tool is an equivalent description by a certain type of di¤erence set relative to disjoint subgroups which we will call a neo-di¤erence set. We then discuss geometric properties and restrictions for the existence of planes of Lenz–Barlotti class I.4. As a side result, we also obtain a new synthetic description of projective triangles in desarguesian planes.