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Imperial College London

Time: 16:00-18:00 (GMT+8), Monday May 15th, 2023

Location: Zoom

**Abstract:**
Let \(G\) be a group acting transitively on a finite set \(S\). Then \(G\) acts on \(S\times S\) component wise.
Define the orbitals to be the orbits of \(G\) on \(S\times S\). 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.

Host: 丁兆宸 Zhaochen Ding