The doubly transitive flocks of quadratic cones (Q1372612)

In this paper, the authors classify all flocks of a quadratic cone in \(\text{PG} (3,q)\), \(q\) odd, admitting a doubly transitive automorphism group. In fact, by earlier results, it is enough to classify the doubly transitive semifield flocks. It turns out that only the Kantor flocks have that property. This follows from a more general classification of semifield flocks satisfying some strong divisibility condition (but satisfied in the doubly transitive case). Together with some earlier results of the authors [Geom. Dedicata 61, No. 1, 71-85 (1996; Zbl 0868.51013)], this gives the following theorem: A flock of an oval or quadratic cone in \(\text{PG} (3,q)\) admitting a doubly transitive automorphism group in \(\text{PGL}_4 (q)\) is either a translation oval flock of Thas \((q\) even), a Kantor flock \((q\) odd), a Fisher-Thas-Walker flock, or a linear flock (all \(q)\). The proof uses the associated spreads and translation planes.