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Category Archives: doing mathematics
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
Graphs on groups, 11
A brief interlude to describe another recent preprint, and as in the preceding post I will concentrate on one result in the paper. I don’t know why it happens, but in this project one of the most interesting graph parameters … Continue reading
Posted in doing mathematics, exposition
Tagged clique number, nilpotent group, solubility graph, souble group
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Graphs on groups, 10
The lesson of this post and the next in the series is that the most interesting questions (to me, anyway) are not about the girth of the deep commuting graph but instead about the classes of groups G defined by … Continue reading
Posted in doing mathematics, exposition
Tagged 2Engel group, commuting graph, conjugacy, Dedekind group, enhanced power graph, power graph
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Graphs on groups, 9
We continue to make progress with the graphs on groups project, but this post attempts to step back and look at the whole thing. What use is all this? Once, after I talked at a departmental colloquium at the University … Continue reading
Posted in doing mathematics, exposition, open problems
Tagged graph theory, group theory, number theory, open problems
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