# 群与图讨论班 Seminars on Groups and Graphs

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# The orbital diameter of primitive permutation groups

## Kamilla Rekvényi

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

Slides