Applications of Group Theory to Finite Geometry

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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.