Abstract

This paper describes a complex of related ideas, ranging from Urbanik’s $v^{\ast }$-algebras, through Deza’s geometric groups and Zilber’s homogeneous geometries, to Sims’ bases for permutation groups and their use in defining “size” parameters on finite groups, with a brief look at Cherlin’s relational complexity. It is not a complete survey of any of these topics, but aims to describe the links between them.

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