A new characterization of the desarguesian and the Figueroa plane

Abstract

A result of Dempwolff [4] asserts that a projective plane Π of order q3 admitting G≅PGL(3,q) as a collineation group contains a G-invariant subplane π0 isomorphic to PG(2,q) on which G acts 2-transitively. Moreover, G splits the point set and the line set of Π into 3 orbits Pi(Π) and Li(Π), i=1,2,3, consisting of the points (respectively lines) of Π incident with q+1, 1 or 0 lines (respectively points) of π0. In this paper it is proven that Π is either the desarguesian or the Figueroa plane of order q3 if, and only if, (P3(Π),L3(Π)) is isomorphic to (P3(D),L3(D)), where D≅PG(2,q3).