Generalized Quadrangles and Transitive Pseudo‐Hyperovals

A pseudo‐hyperoval of a projective spacePG ( 3 n − 1 , q ) , q even, is a set ofq 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 stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles Q that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that Q is flag‐transitive and isomorphic toT 2 * ( H ) , where H is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).