Nonexistence of generalized quadrangles admitting a point-primitive and line-primitive automorphism group with socle \(\textrm{PSU}(3,q)\), \(q\ge 3\)

A central problem in the study of generalized quadrangles is to classify finite generalized quadrangles satisfying certain symmetry conditions. It is known that an automorphism group of a finite thick generalized quadrangle \(\mathcal {S}\) $\mathcal{S}$ acting primitively on both the points and lines of \(\mathcal {S}\) $\mathcal{S}$ must be almost simple. In this paper, we initiate the study of finite generalized quadrangles admitting a point-primitive and line-primitive automorphism group with socle being a unitary group. We develop a group-theoretic tool to prove that the socle of such a group cannot be \(\textrm{PSU}(3,q)\) $\text{PSU}(3,q)$ with \(q\ge 3\) $q\ge 3$ .