Peter Cameron's homepage

Welcome to my new homepage on GitHub. Under construction 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

On this site



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:

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. SCImago Journal & Country Rank

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

Page revised 9 March 2024

A problem

Cayley's Theorem in group theory states that every group of order n can be embedded as a subgroup of the symmetric group Sn. 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 pm, we found a lower bound of pcm2 and an upper bound roughly ppm.

Subproblem 1: is there an upper bound which is pmc for some c?

We were able to find the exact order of the smallest abelian group containing every abelian group of order pm; it is pF(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 non-abelian group can contain all abelian groups of order pm?

Old problems are kept here.