Peter Cameron's homepage

Welcome to my new homepage on GitHub. This page is under construction (and probably always will be!)

I am a quarter-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 zbMATH profile, MathSciNet profile, Google Scholar ORCID record: orcid.org/0000-0003-3130-9505 Mathematical genealogy, Wikipedia page CV and publications (PDF) Biographical stuff, travel diaries On this site 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) Problems from my QMUL homepage

Elsewhere My homepage at QMUL Papers on arXiv Algebra and Combinatorics at St Andrews St Andrews pure mathematics seminars CIRCA ORCiD Lecture notes Australasian Journal of Combinatorics Cameron Counts Mathematical quotes The Infinite Quest, a short course from the IAI Academy Mathematics: the next generation (LMS–Gresham lecture)

Research snapshots

Generalised wreath products

Let M be a set carrying a partial order, and (G(m)) be a family of transitive permutation groups indexed by M. The generalised wreath product of the family over the poset is a construction of a permutation group on the cartesian product of the domains; it is too complicated to define here, but note that the g.w.p. over a 2-element antichain is the direct product, while the g.w.p. over a 2-element chain is the wreath product.

In a joint paper with Marina Anagnostopoulou-Merkouri and Rosemary Bailey, we prove a number of results. Two of them are:

• a generalisation of the Krasner–Kaloujnine theorem: a transitive group which leaves invariant a distributive lattice of commuting equivalence relations is embeddable in a generalised wreath product of groups which can be extracted from the given action;
• the generalised wreath product over an arbitrary poset is the intersection of the iterated wreath products of the same groups over all linear extensions of the poset.

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.

A problem

Richard Parker, who died recently, made a beautiful observation about finite permutation groups. He showed that the number P_k of orbits of a permutation group G on the set of k-cycles of elements of the symmetric group which actually occur in elements of G is the average, over G, of the number of points lying in k-cycles of an element of G. As a consequence, this result generalises the Orbit-Counting Lemma, and also shows that the sum over k of the numbers P_k is equal to the degree of the permutation group. The tuple (P₁,…,P_n) is called the Parker vector of G.

Daniele Gewurz showed me some time ago that the Parker vectors of the two wreath products G wr H and H wr G are equal. This led me to wonder briefly whether the Parker vector of a generalised wreath product depends only on the factors and not on the partially ordered set used to glue them together. But this is false in general. The Parker vector of C₂ wr C₂ is (1,2,0,1), whereas the Parker vector of C₂ × C₂ is (1,3,0,0).

Problem: Is it true that, if the factors of a generalized wreath product have pairwise coprime orders, then the Parker vector is independent of the poset used in the construction but depends only on the factors?

Old poblems are kept here.