On rank 4 projective planes

Abstract

Let a finite projective plane be called rank m plane if it admits a collineation group G of rank m, let it be called strong rank m plane if moreover GP=G1 for some point-line pair (P,1). It is well known that every rank 2 plane is desarguesian (Theorem of Ostrom and Wagner). It is conjectured that the only rank 3 plane is the plane of order 2. By [1] and [7] the only strong rank 3 plane is the plane of order 2. In this paper it is proved that no strong rank 4 plane exists.