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 British Combinatorial Committee, BCC Conference List St Andrews pure mathematics seminars 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

Low in the permutation group hierarchy

With Marina Anagnostopoulou-Merkouri and Enoch Suleiman, I defined a concept for finite permutation groups lying between transitivity and primitivity, which we called "pre-primitivity". It has the property that it is logically independent of quasiprimitivity, but together with quasiprimitivity it is equivalent to primitivity. So it throws some light on the gap (which confused Galois, who devised both concepts) between quasiprimitivity and primitivity. We proved various things about pre-primitivity. The paper is here.

In subsequent work with Marina, I have defined an even weaker concept, which we don't have a name for yet, tentatively the OBS property. The term comes from experimental design in statistics, where an orthogonal block structure is a set of partitions of a finite set which forms a lattice (with top element the partition with a single part and bottom element the partition into singletons), such that any partition in the set is uniform, and any two commute (as relations). A transitive permutation group has the OBS property if the partitions invariant under the group form an orthogonal block structure. This property lies between transitivity and pre-primitivity, and almost all transitive groups appear to satisfy it. This is still under investigation.

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

A (total) ordering of a finite vector space is called compatible if every order-preserving bijection between subspaces of the same dimension is linear.

It is straightforward to show that every finite vector space has a compatible ordering: for example, choose any ordering of the finite field F; then the lexicographic ordering of Fⁿ is compatible, for any n.

Problem: How many compatible orderings does the n-dimensional vector space over the field of order q have? (Formulae are known for the cases n = 1 and n = 2.)

Old problems are kept here.