Groups of hyperovals in Desarguesian planes (Q1022864)

The authors prove a number of results about the automorphism groups of hyperovals in Desarguesian planes, paying special attention to hyperovals admitting insoluble groups of automorphisms. The main result shows that if \({\mathcal H}\) is a hyperoval with an insoluble group of automorphisms \(G \subseteq P\Gamma L(3,q)\), \(q \geq 4\), then \(G\) fixes a subplane \(\pi_0\) of order \(q_0\), \(\pi_0 \cap {\mathcal H}\) is a regular hyperoval of \(\pi_0\), and \(G\) has a normal subgroup isomorphic to \(\text{PGL}(2,q_0)\). They further show that a hyperoval of \(\text{PG}(2,q)\) with homography stabilizer of order greater than \(3(q-1)\) is a translation hyperoval. Finally, they use computer software to verify that the homography group of a Cherowitzo hyperoval is trivial for \(q>8\).