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arXiv · 1406.6445

Generalised quadrangles and transitive pseudo-hyperovals

John Bamberg·Stephen P. Glasby·Tomasz Popiel·Cheryl E. Praeger
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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 $\PG(2,16)$.

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John Bamberg, Stephen P. Glasby, Tomasz Popiel, Cheryl E. Praeger. 2014-06-25. Generalised quadrangles and transitive pseudo-hyperovals. https://arxiv.org/abs/1406.6445

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