Peter Cameron's homepage

Welcome to my new homepage on GitHub. 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. 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, ADE: Patterns in Mathematics, The Shrikhande graph (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 colloquia 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

The Frobenius–Wielandt theorem

The famous theorem of Frobenius can be stated in two different-looking ways:

• In a transitive (but not regular) finite permutation group in which the non-identity elements fix at most two points, the identity and fixed-point-free elements form a normal subgroup.
• If the finite group G has a non-trivial proper subgroup H such that the intersection of H and g⁻¹Hg is trivial for any element g not in H, then the identity and the set of elements not in conjugates of H form a normal subgroup.

This led to two rather different extensions of the theorem. On the permutation group side, Zantema (in connection with a number theory problem) examined the fixed-point-free elements in an arbitrary transitive group, and this was taken further by Bailey, Giudici, Royle and me who posed the following problem. If D(G) is the subgroup generated by fixed-point-free elements, what can the "derangement quotient" G/D(G) be? Clearly any Frobenius complement can occur, but we found some other examples.

On the abstract group side, Wielandt generalised Frobenius' theorem to a class of groups now called Frobenius–Wielandt groups, and Scoppola examined which groups of prime power order could occur as "Frobenius–Wielandt complements".

We have now joined forces and shown that the two approaches are more-or-less equivalent; each throws more light on the other, and in particular, Scoppola's results give further examples of derangement quotients.

The paper by Bailey, Gavioli, Scoppola and me is here.

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 beautiful theorem of Bettina Eick shows that a finite group G is isomorphic to the Frattini subgroup of a finite group if and only if its inner automorphism group is contained in the Frattini subgroup of its automorphism group.

Consider the following question: Given a finite group G, is it isomorphic to the derived group of a finite group? Embarrassingly, we don't know if this question is even decidable!

Problem: Show that the above question is decidable and give an algorithm to decide it.

Old poblems are kept here.