The Uniqueness of the Generalized Quadrangle of Order $5$ with an Axis of Symmetry

Abstract

We show that every generalized quadrangle of order $(4,6)$ with a spread of symmetry is isomorphic to the Ahrens-Szekeres generalized quadrangle $\text{AS}(5)$. It then easily follows that every generalized quadrangle of order $5$ with an axis of symmetry is isomorphic to the classical generalized quadrangle $\text{Q}(4,5)$.