Translation planes of order $q²$ admitting collineation groups of order $q³u$ preserving a parabolic unital

Abstract

The set of translation planes of order $q²$ that admit collineation groups of order $q³u$, where u is a prime p-primitive divisor of $q²-1$, consists of exactly the Desarguesian plane, assuming that the group does not contain a translation subgroup of order a multiple of $q²$. This applies to show that if the group preserves a parabolic unital then the plane is forced to be Desarguesian.