On rank 5 projective planes

Abstract

In this paper we continue the study of projective planes which admit collineation groups of low rank (Kallaher [1] and Bachmann [2,3]). A rank 5 collineation group of a projective plane ℙ of order n≠3 is proved to be flag-transitive. As in the rank 3 and rank 4 case this implies that is ℙ not desarguesian and that n is (a prime power) of the form m4 if m is odd and n=m2 with m≡0mod4 if n is even. Our proof relies on the classification of all doubly transitive groups of finite degree (which follows from the classification of all finite simple groups).

Keywords

• projective planes
• rank 5 groups.