Doubly Transitive Groups

Let U be a hyperbolic unital of Buekenhout type embedded in a translation plane pi of order q2 with spread in PG(3, q). That is to say that U arises from a quadric in PG(4, q) and in this setting, U has q + 1 points on the line at infinity that form the infinite points of a regulus net with the translation plane. It is shown in Johnson and Pomareda [804] that any collineation subgroup of SL(2, q)◦ SL(2, q) of pi that leaves invariant the regulus net can be arranged to leave invariant the hyperbolic unital.