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.
On this site
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.
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.
Diagonal groups are one class of primitive permutation groups arising in the O'Nan–Scott theorem, and are slightly mysterious. With Rosemary Bailey, Cheryl Praeger and Csaba Schneider, I am working on understanding these groups better. As spin-offs, I have shown (with John Bray, Qi Cai, Pablo Spiga and Hua Zhang), using the Hall–Paige conjecture, that diagonal groups with at least three simple factors in the socle are non-synchronizing (arXiv 1811.12671), and (with Sean Eberhard) that they preserve non-trivial association schemes (arXiv 1905.06509).
The Sylvester graph is a remarkable distance-transitive graph of valency 5 on 36 vertices, which can be constructed using the even more remarkable outer automorphism of the symmetric group S6, as shown by Sylvester (and described in Chapter 6 of my book with Jack van Lint). Now there is no affine plane of order 6; but, with Rosemary Bailey, Leonard Soicher and Emlyn Williams, I have been investigating block designs for which there is a Sylvester graph on the set of treatments, and the concurrences are 2 for edges of the graph and 1 for all other pairs. They, and designs obtained by deleting resolution classes, are good substitutes for the missing lattice designs. There seem to be many such designs; but just one admits all the symmetries of the Sylvester graph.
Old research snapshots are kept here.
School of Mathematics and Statistics
University of St Andrews
St Andrews, Fife KY16 9SS
Tel.: +44 (0)1334 463769
Fax: +44 (0)1334 46 3748
[oops – wrong saint!]
Page revised 3 June 2019
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.