Abstract

We show that if a hyperoval $\mathcal{\mathscr{H}}$ of $PG2q$, $q>4$, admits an insoluble group $G$, then $G$ fixes a subplane ${\pi }_0$ of order $q_0>2$, $\mathcal{\mathscr{H}}$ meets ${\pi }_0$ in a regular hyperoval of ${\pi }_0$ on which $G\cap PGL3q$ induces $PGL2q_0$, and if $\mathcal{\mathscr{H}}$ is not regular then $q>q_0^2$. We also bound above the order of the homography stabilizer of a non-translation hyperoval of $PG2q$ by $3\left(q-1\right)$. Finally, we show that the homography stabilizer of the Cherowitzo hyperovals is trivial for $q>8$.

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Mathematical Subject Classification 2000

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