Nick Gill

A graph connected to the conjugacy classes of a group and an application to group actions

Pierre Guillot, Martin Liebeck and I have recently started studying the following graph: start with a conjugacy class of involutions, C, inside a group G. Define a graph Gr(C) whose vertices are the involutions in C with two vertices, x and y, joined by an edge if the product x.y is also in C. I will, first, present some results that describe the connected components of the graph Gr(C) for G a member of various families of finite simple group. I will, second, d, describe how information about the graph Gr(C) can be used to classify the “binary actions” of the group G. (The definition of a “binary action” has its roots in the model theory of Gregory Cherlin – I will define binary actions in the talk!)