### The orbital diameter of primitive permutation groups

Algebra Seminar

15th February 2023, 2:30 pm – 3:30 pm

Fry Building, 2.04

Let G be a group acting transitively on a finite set S. Then G acts on SxS component wise. Define the orbitals to be the orbits of G on SxS. The diagonal orbital is the orbital of the form D = {(a,a) : a in S}. The others are called non-diagonal orbitals. Let O be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set S and edges given by the pairs (a,b) in O. If the action of G on S is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding explicit bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups.

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