Generalised quadrangles and transitive pseudo-hyperovals

A pseudo-hyperoval of a projective space $\text{PG}(3n-1,q)$, $q$ even, is a set of $q^n+2$ subspaces of dimension $n-1$ such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is elementary. We then deduce from this result a classification of the thick generalised quadrangles $\mathcal{Q}$ that admit a point-primitive, line-transitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that $\mathcal{Q}$ is flag-transitive and isomorphic to $T_2^{\ast }(\mathcal{H})$, where $\mathcal{H}$ is either the regular hyperoval of $\text{PG}(2,4)$ or the Lunelli–Sce hyperoval of $\text{PG}(2,16)$.