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This page is under
construction (and probably always will be!)
I am an Emeritus 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.
I am a Fellow of the Royal Society of Edinburgh.
About me
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Let A be a magma, a set with a binary operation. A complete mapping for A is a bijection θ on A such that the map ψ defined by ψ(a) = a·θ(a) is also a bijection, where · is the operation. It defines a transversal of the Cayley table of the magma, a set of cells meeting each row, column and symbol in a unique element.
The celebrated Hall–Paige conjecture asserted that a finite group has a complete mapping if and only if either it has odd order or its Sylow 2-subgroups are non-cyclic. They proved the necessity of the conditions, but sufficiency had to wait for the work of Wilcox, Evans and Bray in 2009.
Little can be said about the existence of complete mappings for arbitrary finite magmas. One simple necessary condition is that every element of the magma can be expressed as a product; more generally, the set of products with fixed left factors has a transversal, and similarly for right factors.
A bit less than ten years ago, João Araújo asked me what could be said about semigroups with complete mappings. It has taken a while, and required the accumulation of three more coauthors (Wolfram Bentz, Kevin Hendrey and Michael Kinyon), but we now have a workable answer to the question; it is too complicated to state here, but I note that it uses the Green–Rees theory of principal factors with a variety of combinatorial tools such as unimodular matrices, Hall's marriage theorm, the proof of the permanent conjecture, and Bevis–Hall–Katz incidence theory over finite abelian groups.
One consequence of our result, with quite a bit more work, is that the complete transformation semigroup Tn has a complete mapping for exactly the same values of n as the symmetric group Sn does, namely, all n different from 2 and 3.
You can find the paper on the arXiv at 2608.25092.
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 2026 |
Consider the following three graphs defined on a finite group G:
It is known that the power graph and enhanced power graph are equal if and only if every element of G has prime power order. After preliminary work by Higman and Suzuki, such groups were all determined by Brandl.
Problem: For which groups are the power graph and the intersection power graph equal?
It is known that groups in which all elements have prime order have this property, and that it implies that the power graph is a cograph and a chordal graph. But an exact characterisation is unknown.
Most of what we know about this question is in my paper with Sudip Bera, Discrete Math. 349 (2026), paper 115387; doi: 10.1016/j.disc.2026.115387.
Old poblems are kept here.
orcid.org/0000-0003-3130-9505