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Category Archives: exposition
London Combinatorics Colloquia 2023
In London last week for this (usually) annual event, now happily back live. In contrast to usual practice, I will discuss the second day first. Norman Biggs was the founder of the combinatorics group at the London School of Economics, … Continue reading
Bases
A base for a permutation group is a sequence (a1,…,ar) of points in the permutation domain whose pointwise stabiliser is the identity. Bases were introduced in computational group theory by Charles Sims, since two elements of a permutation group are … Continue reading
Posted in doing mathematics, exposition
Tagged bases, irredundant bases, minimal bases, Mornington Crescent, permutation groups
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Orthogonal block structures
If you google “commuting equivalence relations” you will find signs of a large and rich literature. I want to talk about one aspect, on the border of algebraic combinatorics and experimental design: orthogonal block structures. Look up orthogonal block structures … Continue reading
Posted in doing mathematics, exposition
Tagged association scheme, partition lattice, permutation group, statistics
1 Comment
Between transitive and primitive
Many familiar properties of permutation groups form a hierarchy. A few years ago, with João Araújo and Ben Steinberg, I wrote a survey on properties related to synchronization of finite automota, lying between primitivity and 2transitivity. Here is a new … Continue reading
Posted in doing mathematics, exposition
Tagged permutation groups, primitive, quasiprimitive, transitive
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Mathematical Structures
My last major job at Queen Mary University of London more than ten years ago was designing and presenting a new firstsemester firstyear module to be taken by all students on mathematics programmes or joint programmes involving mathematics. I discussed … Continue reading
Two things that should be related
Here are two things that look as if there should be a relation between them. A graph duality In my paper on graphs defined on groups, I invented an ad hoc duality relation between pairs of graphs. The context is … Continue reading
A theorem on polytopes
You know what polygons and polyhedra are. How do we extend their study to higher dimensions? There are two parts to this question. The first involves incidence geometry: vertices, edges, faces, etc. Here the generalisation is fairly straightforward. A polygon … Continue reading
Posted in doing mathematics, exposition
Tagged Dimitri Leemans, Maria Elisa Fernandes, regular polytopes, symmetric group, zoom
3 Comments
The road closure property
My work with João Araújo and other semigroup theorists has produced a number of permutation group properties which lie between primitivity and 2homogeneity, especially the synchronization family. Another of these is the road closure propery, which I have discussed here … Continue reading
BCC29 at Lancaster
Last week we celebrated the 29th British Combinatorial Conference in Lancaster, face to face. (As a side observation, this was by far the largest social gathering I have been at since the start of the pandemic; I found it both … Continue reading
Peter Neumann’s 3p paper
In 1955, Helmut Wielandt published a paper proving the following theorem: Let G be a primitive permutation group of degree 2p, where p is a prime greater than 3, which is not doubly transitive. Then p = 2a2+2a+1 for some positive integer … Continue reading