Generalised quadrangles and transitive pseudo-hyperovals

TL;DR

This paper proves that pseudo-hyperovals with an irreducible transitive stabiliser are elementary and classifies certain automorphism groups of generalised quadrangles, showing they are flag-transitive and related to specific hyperovals.

Contribution

It establishes the elementary nature of pseudo-hyperovals with transitive stabilisers and classifies certain automorphism groups of generalised quadrangles as flag-transitive.

Findings

Pseudo-hyperovals with an irreducible transitive stabiliser are elementary.

Classified thick generalised quadrangles with specific automorphism groups.

Identified isomorphism to quadrangles related to particular hyperovals.

Abstract

A pseudo-hyperoval of a projective space $\PG(3n-1,q)$, $q$ 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 $T_2^*(\mathcal{H})$, where $\mathcal{H}$ is either the regular hyperoval of $\PG(2,4)$ or the Lunelli--Sce hyperoval of $\PG(2,16)$.