Category Archives: open problems

unsolved mathematical problems

Graphs on groups, 9

Posted on 08/11/2021 by Peter Cameron

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 →

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Clive Sinclair

Posted on 18/09/2021 by Peter Cameron

Yesterday I read the news that Clive Sinclair has died. This brought back memories of my first encounter with personal computers nearly 40 years ago. At the time I had a demanding job and three small children, and I was … Continue reading →

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Graphs on groups, 8

Posted on 16/08/2021 by Peter Cameron

The dark clouds seem to have lifted a bit. Perhaps now, that the last rush of conferences for a while is over, life can return to something like normality … For me the most significant event was the last in … Continue reading →

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A small problem

Posted on 26/07/2021 by Peter Cameron

In connection with the research discussion about graphs and groups, I began to wonder which finite groups have the property that any two elements of the same order are conjugate. I thought about this, and got a certain distance, and … Continue reading →

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A little problem

Posted on 31/05/2021 by Peter Cameron

In connection with the power graphs of unitary groups, I came across the following little number-theoretic conundrum. Can anyone solve it? Let q be an odd power of 2 (bigger than 2). Show that (q2−q+1)/3 is not a prime power … Continue reading →

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A new constant?

Posted on 23/05/2021 by Peter Cameron

This is an appeal for help. Has anyone come across the constant 2.648102…? Here is the background, which connects with my previous posts about graphs on groups. We are interested in the clique number of the power graph of the … Continue reading →

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Graphs on groups, 5

Posted on 26/03/2021 by Peter Cameron

I gave two lectures on this stuff to a new research seminar on Groups and Graphs, run by Vijayakumar Ambat in Kochi, Kerala. The first was an introduction to the hierarchy, the second was about cographs and twin reduction, why … Continue reading →

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Graphs on groups, 4

Posted on 18/02/2021 by Peter Cameron

Here is a small problem, mixing group theory and number theory, which might appeal to someone. A couple of definitions. The power graph of a group G has an edge from x to y if one is a power of … Continue reading →

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Induced subgraphs of power and commuting graphs

Posted on 16/12/2020 by Peter Cameron

For those who like thinking about these things, here is a small observation and a few problems. As I have recently discussed, the power graph of a group is perfect. This means that all its induced subgraphs are perfect, and … Continue reading →

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Ramanujan+100

Posted on 15/12/2020 by Peter Cameron

I have just spent the last four days in Kochi, Kerala, at the International Conference on Number Theory and Discrete Mathematics, commemorating Srinivasa Ramanujan, the great Indian mathematician, on the 100th anniversary of his far-too-early death. The conference had perhaps … Continue reading →

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