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Category Archives: doing mathematics
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
COBS and equitable partitions
It happens sometimes that researchers working in different fields study the same thing, give it different names, and don’t realise that there is further work on the subject somewhere else. Here is a story of such a situation, which arose … Continue reading
An apology
What would life be like if I could remember all the things I ever knew? Yesterday I was led to something I posted here twelve years ago. This was based on a talk to the London Algebra Colloquium by Mark … Continue reading
Posted in doing mathematics
Tagged conjugacy supercommuting graph, Jordan's theorem, Mark Wildon
4 Comments
The enhanced power graph is weakly perfect
Earlier this year, I posed a combinatorial problem, a solution to which would imply that, for any finite group G, the enhanced power graph of G is weakly perfect, that is, has clique number equal to chromatic number. Recall that … Continue reading
Posted in doing mathematics
Tagged chromatic number, clique number, enhanced power graph, Euler's totient
19 Comments
More on the 3p paper
I wrote here about Peter Neumann’s paper on primitive permutation groups of degree 3p, where p is a prime number. Well, summer is almost over, but my undergraduate research intern Marina AnagnostopoulouMerkouri and I have done our work and produced … Continue reading
Why I’d like to see this solved
I am aware that quite a number of people have been captivated by the problem I posed. So here is the motivation for it, with some additional remarks and commennts. First, to repeat the problem: Problem: Let n be a … Continue reading
Posted in doing mathematics
Tagged chromatic number, enhanced power graph, GruenbergKegel graph
3 Comments
I’d like to see this solved
Here is a problem that I would really like to see solved. I have spent quite a bit of time on it myself, and have suggested it to a few other people, but it still resists all attacks, though it … Continue reading
Posted in doing mathematics, open problems
9 Comments
Graphs on groups, 13
There are many results about the universality, or otherwise, of various graphs defined on groups: answers to questions of the form “for which graphs Γ is there a group G such that Γ is isomorphic to an induced subgraph of … Continue reading
Graphs on groups, 12
One thing I have learned from the project is that the most interesting question about graphs defined on groups is this: given two types of graph defined on a group G, what is the class of groups for which the … Continue reading
Posted in doing mathematics, exposition
Tagged enhanced power graph, independence graph, power graph, rank, supersoluble group
1 Comment