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