PSL(2,q) as a collineation group of projective planes of small order

Abstract

Let Π be a projective plane of order n admitting a collineation group G≅PSL(2, q) for some prime power q. It is well known for n=q that Π must be Desarguesian. We show that if n<q then only finitely many cases may occur for П, all of which are Desarguesian. We obtain some information in case n=q2 with q odd, notably that G acts irreducibly on П for q≠3, 5, 9.