A generalized quadrangle with an automorphism group acting regularly on the points

We develop several fundamental lemmas on a generalized quadrangle with an automorphism group acting regularly on its set of points. This can be thought of as an improvement of arguments originated in D. Ghinelli [Regular groups on generalized quadrangles and nonabelian difference sets with multiplier −1, Geom. Dedicata 41 (1992) 165–174], not depending on representation theoretic results in U. Ott [Some remarks on representation theory in finite geometries, in: Geometries and Groups, in: Lecture Notes in Math. vol. 893, 1981, pp. 68–110]. Based on them, we show that there is no generalized quadrangle of order ( t 2 , t ) ( t ≥ 2 ) with such an automorphism group.