Peter Cameron's homepage

Welcome to my new homepage on GitHub. This page is under construction (and probably always will be!)

I am a quarter-time Professor in the School of Mathematics and Statistics at the University of St Andrews, and an Emeritus Professor of Mathematics at Queen Mary, University of London. In addition, I am an associate researcher at CEMAT, University of Lisbon, Portugal.

I am a Fellow of the Royal Society of Edinburgh.

Research snapshots

Cauchy numbers

In group theory, Cauchy's Theorem states that, if a prime p divides the order of a finite group G, then G contains a subgroup which is cyclic of order p.

Can we reverse-engineer this? Clearly we cannot ask for a single subgroup, since primes are the only numbers for which such a result would be true. So we say that n is a Cauchy number if there is a finite list F of finite groups such that a finite group G has order divisible by n if and only if G has a subgroup isomorphic to one of the groups in F.

In a paper with David Craven, Hamid Reza Dorbidi, Scott Harper and Benjamin Sambale, I proved that the Cauchy numbers are as follows:
• prime powers;
• 6;
• 2pa, where p is a Fermat prime greater than 3 and a is a positive integer.

For example, the list for n = 6 consists of three groups: the cyclic and dihedral groups of order 6 and the alternating group A4 of order 12.

The paper is here.

Old research snapshots are kept here.

 I am Honorary Editor-In-Chief of the Australasian Journal of Combinatorics, an international open-access journal published by the Combinatorial Mathematics Society of Australasia.

 School of Mathematics and Statistics University of St Andrews North Haugh St Andrews, Fife KY16 9SS SCOTLAND Fax: +44 (0)1334 46 3748 Email: pjc20(at)st-arthurs(dot)ac(dot)uk   [oops – wrong saint!] Page revised 21 August 2024

A problem

Let G be a group. For x,y in G and a natural number n, we define [x, ny] inductively by

• [x1y] = [x, y] = x−1y−1xy,
• [xn+1y] = [[x, ny], y].
The directed Engel graph on G is the graph with vertex set G, with an arc yx if [x, ny] is the identity for some natural number n.

The graph has a loop at each vertex; these loops will not affect what follows.

If G is nilpotent, then any pair of vertices are joined by arcs in both directions, so we have the complete directed graph with loops. It follows from the theorem of Zorn that the converse of this is also true.

The undirected Engel graph is obtained by ignoring directions and loops and replacing a double edge by a single edge.

Problem 1: Does the undirected Engel graph determine the directed Engel graph up to isomorphism? If this is false in general, for which groups does it hold?

Note that the analogous statement holds for the directed and undirected power graph.

In a recent preprint with Rishabh Chakraborty, Rajat Kanti Nath and Deiborlang Nongsiang (arXiv 2408.03879), we showed (among other things) that, if G is a finite soluble group which is not nilpotent, then the directed Engel graph has two vertices joined by a single (non-loop) arc. These vertices can be chosen so that the source of the arc is in the Fitting subgroup of G (which is the set of vertices which send arcs to all vertices) and the target is outside the Fitting subgroup.

Problem 2: Classify the finite soluble groups for which all single arcs in the directed Engel graph have one vertex in the Fitting subgroup.

Old problems are kept here.