Nonexistence of generalized quadrangles admitting a point-primitive and line-primitive automorphism group with socle PSU (3, q), q 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 S acting primitively on both the points and lines of 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 PSU (3, q) with q 3.