Let \(\mathcal {S}\) be a finite thick generalized quadrangle, and suppose that G is an automorphism group of \(\mathcal {S}\). If G acts primitively on both the points and lines of \(\mathcal {S}\), then it is known that G must be almost simple. In this paper, we show that if the socle of G is \(\textrm{PSL}(2,q)\) with \(q\ge 4\), then \(q=9\) and \(\mathcal {S}\) is the unique generalized quadrangle of order 2.

中文翻译：

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在以 $$\textrm{PSL}(2,q)$$ 作为自同构群的有限广义四边形上

设\(\mathcal {S}\)是有限厚广义四边形，并假设G是\(\mathcal {S}\)的自同构群。如果G原始地作用于\(\mathcal {S}\)的点和线，那么已知G必须几乎是简单的。在本文中，我们证明如果G的基是\(\textrm{PSL}(2,q)\)和\(q\ge 4\)，则\(q=9\)和\(\mathcal {S}\)是唯一的 2 阶广义四边形。