Category Archives: open problems

Digraphs on groups

Posted on 27/03/2026 by Peter Cameron

I have spent a lot of time recently thinking about graphs on groups. To recall the rules: the vertex set must be the group (in general, but not here, I allow an automorphism-invariant subset or the quotient by an automorphism-invariant … Continue reading →

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A counting problem

Posted on 08/03/2026 by Peter Cameron

As the tagline for this blog says, I like counting things. Reading my Iran diary reminded me of a counting problem I solved then, of which I am quite proud. But like all good problems, it leaves a loose end, … Continue reading →

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The rational world and the rational Urysohn space

Posted on 03/03/2026 by Peter Cameron

The set Q of rational numbers is obviously an interesting topological space. In 1920, Waclaw Sierpiński gave a lovely characterisation of it. The simplest way to state it is to say that a countable, metrisable, space without isolated points is … Continue reading →

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Why transformation monoids are harder than permutation groups

Posted on 26/11/2025 by Peter Cameron

For permutation groups (or transformation monoids), we don’t need to assume the associative law, since composition of functions is always associative. So a permtation group is a set of mappings satisfying the identity, inverse and closure axioms. This implies that … Continue reading →

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Graphs defined on algebras

Posted on 17/11/2025 by Peter Cameron

Next February, I will be speaking at AAA108 (Arbeitstagung Allgemeine Algebra) in Vienna. I thought this might be a chance to take some of the work about graphs defined on groups, and see whether it can be extended to arbitrary … Continue reading →

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A Catalan question

Posted on 09/07/2025 by Peter Cameron

This week I am nominally at the Permutation Patterns conference. I find myself on the edge of the (very strong) community of Permutation Patterners; also, after a month away, I have lots of catching up to do; also, Bruce Sagan … Continue reading →

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A class of interesting permutation groups

Posted on 27/05/2025 by Peter Cameron

This is the class of primitive groups having an imprimitive subgroup of index 2. Let us make them almost simple as well. For a boring name, I will call this class (P). So I begin with two questions for specialists: … Continue reading →

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A talk by Gareth Jones

Posted on 14/05/2024 by Peter Cameron

Today I attended (remotely) a nice talk by Gareth Jones in the Ural Workshop on Group Theory and Combinatorics, about prime powers in permutation group theory and polynomials taking prime values in number theory. I will give just one example … Continue reading →

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Some challenges on Latin squares

Posted on 06/10/2023 by Peter Cameron

Following the combinatorial design challenges, here are three questions on Latin squares. A Latin square is an n×n matrix with entries from an alphabet of size n (typically the integers from 1 to n) such that each letter appears once … Continue reading →

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Challenges in combinatorial design theory

Posted on 01/10/2023 by Peter Cameron

Since WordPress changed their editor and it is no longer possible to write posts in HTML, I have to find a new solution to posting mathematics. What I have done this time is to put it on my web page … Continue reading →

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