Marina Anagnostopoulou-Merkouri

The regularity number of a simple group

Let G be a transitive permutation group on a set Ω with point stabiliser H. A base for G is a subset of Ω whose pointwise stabiliser in G is trivial. The minimal size of a base is called the base size of G, denoted by b(G), and this classical invariant has been widely studied for many years, finding many applications.
We will see that b(G) coincides with the smallest positive integer k such that G has a regular orbit on the Cartesian product (G/H)^k. As a natural generalisation, we can consider the minimal number r such that G has a regular orbit on (G/H_1) x ... x (G/H_r) for any core-free subgroups H_1,..., H_r of G. We call this invariant the regularity number of G.
In this talk, we will explain how a combination of probabilistic, combinatorial, computational and geometric methods can be used to study the regularity number of finite almost simple and simple algebraic groups, as well as several other related problems. In particular, we will discuss a generalisation of Cameron’s famous base size conjecture and some recent progress on similar problems for simple algebraic groups.
I will report on joint work with Tim Burness.