On finite generalized quadrangles with PSL(2,q) as an automorphism group - Feng, Tao; Lu, Jianbing (Designs, codes, and cryptography, 2023)
Let S be a finite thick generalized quadrangle, and suppose that G is an automorphism group of S . If G acts primitively on both the points and lines of S , then it is known that G must be almost simple. In this paper, we show that if the socle of G is PSL ( 2 , q ) with q ≥ 4 , then q = 9 and S is the unique generalized quadrangle of order 2.