# Bounds for the diameters of orbital graphs of affine groups

GRA2 - Groups, representations and applications: new perspectives

Let $G$ be a permutation group acting on a finite set $X$. An orbital graph of $G$ is a graph with vertex set $X$ whose arc set is an orbit of $G$ on $X \times X$. An orbital graph whose arcs are a subset of the diagonal $\{ (x,x) \mid x \in X \}$ is called a diagonal orbital graph. A famous theorem of Higman states that a transitive permutation group $G$ acting on $X$ is primitive if and only if all non-diagonal orbital graphs are (strongly) connected. A description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the (undirected) diameter of all non-diagonal orbital graphs has been given in a paper by Liebeck, Macpherson, Tent. In this talk we will be interested in diameters of orbital graphs of affine primitive permutation groups. This is joint work with Saveliy V. Skresanov

This talk is part of the Isaac Newton Institute Seminar Series series.