A generalised quadrangle is a point–line incidence geometry $\mathcal{G}$ such that: (i) any two points lie on at most one line, and (ii) given a line $L$ and a point $p$ not incident with $L$, there is a unique point on $L$ collinear with $p$. They are a specific case of the generalised polygons introduced by Tits (1959), and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as automorphism groups. We show that no sporadic almost simple group can act primitively on the points of a finite (thick) generalised quadrangle. We also present two new ideas contributing towards analysing point-primitive groups acting on generalised quadrangles. The first is the outline and implementation of an algorithm for determining whether a given group can act primitively on the points of some generalised quadrangle. The second is the discussion of a conjecture resulting from observations made in the course of this work: any group acting primitively on the points of a generalised quadrangle must either act transitively on lines or have exactly two line-orbits, each containing half of the lines.

甲广义四边形是一个点-线发生几何$\mathcal{G}$ 这样：（i）任何两点最多位于一条线上，并且（ii）给定一条线 $\mathrm{大号}$ 还有一点 $p$ 没事 $\mathrm{大号}$，有一个独特的观点 $\mathrm{大号}$ 与共线 $p$。它们是广义多边形的特殊情况由Tits（1959）提出，这些结构及其自同构群在有限几何中具有某些重要性。理解有限广义四边形的自同构组的一个组成部分是知道哪些组可以在其点上原始地起作用，尤其是，哪些几乎简单的组作为自同构组出现。我们表明，没有零星的几乎简单的组可以对有限（粗）广义四边形的点进行原始操作。我们还提出了两个新的思想，它们有助于分析作用于广义四边形上的点本原组。第一个是算法的概述和实现，该算法用于确定给定组是否可以原始地作用于某个广义四边形的点。