Translation planes of order admitting collineation groups of order preserving a parabolic unital

Abstract

The set of translation planes of order that admit collineation groups of order , where u is a prime p-primitive divisor of , consists of exactly the Desarguesian plane, assuming that the group does not contain a translation subgroup of order a multiple of . This applies to show that if the group preserves a parabolic unital then the plane is forced to be Desarguesian.