Permutation groups, bases and subgroup regularity

Speaker: Tim Burness

Venue: Room 203, Haina Building 2, Zijingang Campus

Abstract: Let G be a transitive permutation group on a finite set X, let H be a point stabiliser and recall that the base size of G, denoted b(G,X), is the minimal size of a base for G. Equivalently, the base size is the minimal integer k such that G has a regular orbit on the Cartesian product (G/H)^k. Seeking a natural generalisation, let us say that a k-tuple (H_1,..., H_k) of core-free subgroups of G is regular if G has a regular orbit on G/H_1 x ... x G/H_k. Then the regularity number of G, denoted R(G), is the minimal integer k such that every k-tuple of core-free subgroups of G is regular. More refined invariants can be defined by imposing additional conditions on the component subgroups, such as solubility or nilpotency, and this leads to natural generalisations of several widely studied conjectures on bases due to Cameron, Pyber and Vdovin.

In this talk, I will introduce the key definitions and I will discuss some of the main methods we use to study subgroup regularity. I will then present some recent results obtained with my PhD student Marina Anagnostopoulou-Merkouri, and I will also discuss work with Hongyi Huang on the regularity of nilpotent subgroups of simple groups, and joint work with Lei Wang on the regularity of irreducible subgroups of classical groups. Along the way, I will also highlight a number of open problems in this area.