Applications of Group Theory to Finite Geometry

群论和几何学相互影响了一个多世纪。几何中最重要的结构是对称的结构，而最重要的群作用于几何。有限几何的最新发展，尽管受到对称性的启发，却使用了最少的群论。该项目旨在通过将广泛的有限群论的结果应用于有限几何中当前的热门话题来纠正这一点。我们的目标是通过找到与目前已知的结构有实质性不同的结构来实现范式的转变。如果它成功了，几何学上的许多活动将随之而来，寻求对这些群论结果的几何解释。由于缺乏对称广义四边形的基本群的特征结果，我们的重点是必要的。

Group theory and geometry have influenced one another for over a century. The most important structures in geometry are the symmetric ones and the most important groups act on geometries. Recent developments in finite geometry, although informed by symmetry, have used a minimum of group theory. The project aims to redress this, by applying results from a broad range of finite group theory to the presently hot topics in finite geometry. Our aim is to achieve a paradigm shift, by finding substantively different structures than those presently known. Should it succeed, much activity in geometry would follow, seeking geometric interpretation of these group theoretic results. Our focus is necessitated by the lack of a result characterising the underlying groups of symmetric generalised quadrangles.