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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
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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. |
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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 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 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.
orcid.org/0000-0003-3130-9505