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, and a teacher at Universidade Aberta, Portugal.
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With João Araujo 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.
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.
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.
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 1 February 2018 
The cycle polynomial F_{G}(x) of a permutation group G is the polynomial ∑{x^{c(g)} : g∈G}, where c(g) is the number of cycles of g (including fixed points).
The orbital chromatic polynomial P_{Γ,G}(x) of the graph Γ and subgroup G of Aut(Γ) is the polynomial whose value at the positive integer x is the number of Gorbits on proper colourings of Γ with x colours.
We call the pair (Γ,G) a reciprocal pair if P_{Γ,G}(x) = (−1)^{n}GF_{G}(−x).
Problem: Find all reciprocal pairs.
See arXiv 1701.06954 for more information.
Old problems are kept here.