Automorphism groups of flocks of oval cones (Q1914052)

The translation oval flocks of J. Thas, which are described in an article by \textit{J. C. Fisher} and \textit{J. A. Thas} [Math. Z. 169, 1-11 (1979; Zbl 0406.51004)], the semifield flocks of Knuth of odd order, and the Betten-Fisher-Thas-Walker flocks for \(q \equiv-1 \pmod 3\) all admit an automorphism group doubly transitive on the planes of the flock. Because of these examples the authors study flocks admitting linear automorphism groups doubly transitive on the planes of the flock. In this article the authors prove that a flock \({\mathcal F}\) of an oval cone in \(\text{PG} (3,q)\), where \(q\) is even, admitting a linear automorphism group doubly transitive on the planes of \({\mathcal F}\) is either linear or a translation oval flock of Thas. The odd order case will be considered in another article. The proof of this result uses the work of the authors [J. Comb. Theory, Ser. A 73, No. 1, 60-76, Art. No. 0002 (1996; Zbl 0856.51003)] on rigid groups of flocks. An interesting result obtained by the authors says the translation planes associated with nonlinear flocks of quadratic cones in \(\text{PG} (3,2^r)\) cannot have Baer involutions.