Welcome to my new homepage on GitHub. This page is under construction (and probably always will be!)
I am a quartertime 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
On this site

Elsewhere

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 reverseengineer 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:For example, the list for n = 6 consists of three groups: the cyclic and dihedral groups of order 6 and the alternating group A_{4} of order 12.
The paper is here.
Old research snapshots are kept here.
I am Honorary EditorInChief of the Australasian Journal of Combinatorics, an international openaccess 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)starthurs(dot)ac(dot)uk [oops – wrong saint!] 
Page revised 9 March 2024 
Cayley's Theorem in group theory states that every group of order n can be embedded as a subgroup of the symmetric group S_{n}. But usually this group is much too large.
Problem: What is the size F(n) of the smallest group which embeds every group of order n?
In a paper with David Craven, Hamid Reza Dorbidi, Scott Harper and Benjamin Sambale, we looked at this problem, without a lot of success. For the case where n is a prime power p^{m}, we found a lower bound of p^{cm2} and an upper bound roughly p^{pm}.
Subproblem 1: is there an upper bound which is p^{mc} for some c?
We were able to find the exact order of the smallest abelian group containing every abelian group of order p^{m}; it is p^{F(m)}, where
F(m) = Σ⌊m/k⌋,
where the sum over k runs from 1 to m. Dirichlet found the order of magnitude of this function; it is roughly m log m.
Subproblem 2: Is it true that no smaller nonabelian group can contain all abelian groups of order p^{m}?
Old problems are kept here.