# Graphs and groups

Friday 7th December 2018, Lancaster University

Local organiser: Nadia Mazza

This is the sixth meeting of the Research Group Functor Categories for Groups (FCG). We shall focus on the interplay of graphs and groups, and how each of these structures can be used in the study of the other. Groups act on varied combinatorial structures and graphs in particular. On the other

hand, given an abstract group, there are several ways to construct a graph associated to the group, such as π-products involution graphs, or

commuting graphs. Recently, the approach of studying groups via associated graphs has been extended to hypergraphs, which leads to a useful generalisation of the theory.

The provisional timetable is as follows:

**1:15-2:15pm:**Jason Semeraro (University of Leicester),*Computing with fusion systems***2:25-3:25pm:**Nayab Khalid (University of St Andrews),*Conjugacy and Dynamics in Rearrangement Groups***3:25-3:55pm:**Tea and coffee**3:55-4:55pm:**Peter Rowley (University of Manchester),*Chamber Graphs, GAB's and Other Fantastic Beasts*

All talks will be held in the *Postgraduate Statistics Centre Lecture Theatre* (Room A54 in the encircled building on the campus map).

To register for the event, please email the local organiser Dr Nadia Mazza (n.mazza@lancaster.ac.uk).

The FCG Research Group is supported by an LMS Joint Research Groups in the UK Scheme 3 grant. Limited funding is available for PhD students, allocated on a first come first served basis.

For UK-based mathematicians with caring duties the LMS has a Caring Supplementary Grant scheme which allows participants of meetings like ours to apply for help covering caring costs.

## Abstracts

**Jason Semeraro** (University of Leicester), *Computing with fusion systems*

This is joint work with Chris Parker.

Let p be a prime and S be a finite p-group. A saturated fusion system on S is a small category whose objects are subgroups of S and whose morphisms satisfy some relations motivated by G-conjugacy relations which hold whenever there is a finite group G with Sylow p-subgroup S. There is a notion of a "normal subsystem" of a fusion system, and therefore a notion of "simple fusion system." A simple fusion system is p-locally indistinguishable from a finite simple group. Fusion systems which do not arise from groups are called “exotic”.

In an attempt to better understand how fusion systems arise, and the apparent ubiquity of exotic fusion systems at odd primes, we develop software which will list all simple fusion systems on S. In order to prove a fusion system F is saturated we use its presentation to calculate a "fusion graph" which allows one to list all morphism sets in the category. We use our programs to calculate all fusion systems on groups of order p^n for (p,n) in {(3,4),(3,5),(3,6),(5,4),(5,5),(7,4),(7,5)}. Apart from presenting our results, in this talk I’ll discuss the theoretical underpinning of our algorithms

**Nayab Khalid** (University of St Andrews), *Conjugacy and Dynamics in Rearrangement Groups*

Rearrangement groups (groups of homeomorphisms) of fractals (self-similar topological spaces) were introduced by Belk and Forrest in 2016. We are interested in how the behaviour of these groups depends on the geometric structure of the space. We have found that these groups have an infinite generating set that corresponds to the basic open sets of the topological space. In this talk, I will show you how, once we have the right generating set, all dynamical behaviour of these groups can be described by a few conjugacy relations.