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.

About me My official page My diary zbMATH profile, MathSciNet profile, Google Scholar ORCID record: orcid.org/0000-0003-3130-9505 Mathematical genealogy, Wikipedia page CV and publications (PDF) Biographical stuff, travel diaries On this site Talks, publications, conjectures, coauthors, students Books: Introduction to Algebra, Combinatorics, Notes on Counting, Permutation Groups, Sets, Logic and Categories (some dead links to be sorted out) Problems from my QMUL homepage

Elsewhere My homepage at QMUL Papers on arXiv Algebra and Combinatorics at St Andrews St Andrews pure mathematics seminars CIRCA ORCiD Lecture notes Australasian Journal of Combinatorics Cameron Counts Mathematical quotes The Infinite Quest, a short course from the IAI Academy Mathematics: the next generation (LMS–Gresham lecture)

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.

• prime powers;
• 6;
• 2p^a, 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.

A problem

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

• [x, ₁y] = [x, y] = x⁻¹y⁻¹xy,
• [x, _n+1y] = [[x, _ny], y].

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.