Welcome to my St Andrews homepage. This page is under construction (and probably always will be!)
I am a halftime 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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The permutation group G has the kexistential transversal property, or ket for short, if there is a ksubset A of the domain (the witnessing set) such that, for any kpartition 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 ket 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.
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.
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 torsionfree groups, the directions are indeed determined. See arXiv 1705.01586.
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 spinoffs, 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 nonsynchronizing (arXiv 1811.12671), and (with Sean Eberhard) that they preserve nontrivial association schemes (arXiv 1905.06509).
School of Mathematics and Statistics
University of St Andrews North Haugh St Andrews, Fife KY16 9SS SCOTLAND 
Tel.: +44 (0)1334 463769 Fax: +44 (0)1334 46 3748 Email: pjc20(at)starthurs(dot)ac(dot)uk [oops – wrong saint!] 
Page revised 21 May 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 Ginvariant 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.