Two-transitive groups on a hyperbolic unital (Q2426429)

A unital \(\mathcal{U}\) in a projective plane of order \(q^2\) is a set of \(q^3+1\) points such that the lines of the plane are incident with exactly one point, or exactly \(q+1\) points of \(\mathcal{U}\). As the main result of the present paper, the authors prove that if the projective extension of a translation plane of order \(q^2\) contains a unital \(\mathcal{U}\) and admits a collineation group that leaves \(\mathcal{U}\) and a secant line \(L\) invariant, and acts two-transitively on \(L\cap\mathcal{U}\), then the plane is Desarguesian and the unital is classical. ('Classical' means that it may be obtained by Buekenhout's construction.) This paper is the last in a four-term series on doubly transitive ovals and unitals, and the main results of the previous papers are used effectively here.