Peter Cameron's homepage

Welcome to my St Andrews homepage. This page is under construction (and probably always will be!)

I am a half-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 and timetable MathSciNet profile, Google Scholar ORCID record: orcid.org/0000-0003-3130-9505 Mathematical genealogy, Wikipedia page CV and publications (PDF) Short bio, some interviews On this site Teaching pages: MT5821, Advanced Combinatorics 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)

Elsewhere My homepage at QMUL Papers on arXiv, or with abstracts (Atom feed) Algebra and Combinatorics at St Andrews British Combinatorial Committee, BCC Conference List St Andrews seminars: Pure Maths, Algebra/Combinatorics Lecture notes Cameron Counts Walks page, Travel diaries Mathematical quotes The Infinite Quest, a short course from the IAI Academy Mathematics: the next generation (LMS–Gresham lecture)

Research snapshots

The existential transversal property

The permutation group G has the k-existential transversal property, or k-et for short, if there is a k-subset A of the domain (the witnessing set) such that, for any k-partition P of the domain, there exists g∈G such that Ag is a transversal to P. With João Araújo and Wolfram Bentz, I determined all the finite permutation groups of degree n with the k-et property for 4 ≤ k ≤ n/2, apart from a few exceptional cases with k = 4 or k = 5. There are applications to regular semigroups. See arXiv 1808.06085.

Integrable groups

With João Araújo, Carlo Casolo and Francesco Matucci, I determined the set of natural numbers n with the property that every group of order n is the derived subgroup of some group. This set is closely related to (but slightly larger than) the set of n for which every group of order n is abelian. We also showed that if a finite group is a derived group, then it is the derived group of a finite group. See arXiv 1803.10179.

Equitable partitions of Latin square graphs

With Rosemary Bailey, Alexander Gavrilyuk, and Sergey Goryainov, I determined the equitable partitions of Latin square graphs under an extra assumption on eigenvalues. A partition is equitable if, given any two parts, the number of neighbours in the second part of a vertex in the first depends only on the parts and not on the chosen vertex. See arXiv 1802.01001.

Power graphs of torsion-free groups

In the directed power graph of a group, there is an arc from x to y if y is a power of x; for the undirected power graph, just ignore directions. The power graph does not in general determine the directed power graph, even up to isomorphism (though it does for finite groups). With Horacio Guerra and Šimon Jurina, I showed that for various torsion-free groups, the directions are indeed determined. See arXiv 1705.01586.

A problem

Let G be a primitive permutation group with a regular subgroup H; then we can identify the set of points permuted with H so that H acts on itself by right multiplication. In particular, any G-invariant graph is a Cayley graph for H. Suppose that Γ is such a graph (not complete or null), and A a clique and B an independent set in Γ such that |A|·|B| = |H|.

Problem. Is it true that A⁻¹ is also a clique?

This is true if G contains also H acting by left multiplication, so in particular if H is abelian. Its truth in general would resolve a problem in synchronization theory.

Old problems are kept here.