Permutation groups, transitive subgroups and bases

In this talk, we will discuss two major themes in permutation group theory, which have been the focus of intense research in recent years.

There has been great interest for many decades in understanding the transitive subgroups of permutation groups. This long-standing problem has a close connection with group factorisations, which continues to be a very active area of research. Building on the recent and remarkable classification of factorisations of almost simple groups, I will present recent joint work with Lei Wang on the regular subgroups and the soluble transitive subgroups of primitive groups.

In a different direction, let us recall that a base for a permutation group on a set X is a subset of X with trivial pointwise stabiliser. Bases arise naturally in many different settings, finding a wide range of applications. Here I will highlight their connection with some interesting generation properties of finite simple groups, and I will report on my recent work on bases for primitive groups of minimal size.