Category Archives: doing mathematics

how is it done?

Graphs defined on groups

Posted on 04/12/2020 by

Apologies; I have been so busy lately that very little has got written up. Let me try to remedy this with a quick tour through some recent mathematical developments. As some of my posts have hinted, one topic I have … Continue reading →

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A paradox, and where it led

Posted on 19/11/2020 by

What is the difference between a contradiction and a paradox? A contradiction is a dead end, a sign that the road leads nowhere and you should turn back and take the other road. A paradox, however, is an invitation to … Continue reading →

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Perfectness of the power graph

Posted on 04/10/2020 by

The power graph of a group is the graph whose vertices are the group elements (sometimes the identity is excluded but it doesn’t matter here), in which x and y are joined if one is a power of the other. … Continue reading →

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On the Frattini subgroup

Posted on 07/09/2020 by

I wrote earlier about the Frattini subgroup of a group. It can be defined in either of two ways (as the set of non-generators of a group, the elements which can be dropped from any generating set containing them; or … Continue reading →

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Surprising fun fact

Posted on 06/09/2020 by

I have just found a proof of the following. Usual caveat: nobody else has read the proof yet, and I have not carefully checked it. Let G be a finite group. The finite group H will be called an inverse … Continue reading →

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Integrals of groups revisited

Posted on 04/09/2020 by

After my trip to Florence in February, I wrote about the work I did there with Carlo Casolo and Francesco Matucci. After Carlo’s untimely death the following month, we were left with many pages of notes from him about the … Continue reading →

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A problem

Posted on 19/08/2020 by

I seem to have too many balls in the air at the moment. So let me drop one here. Any thoughts very welcome. A k-hypergraph consists of a set X of vertices and a collection of k-element subsets called edges. … Continue reading →

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A puzzle

Posted on 05/08/2020 by

What is the sequence that begins like this? 0, 1, 2, 4, 5, 8, 12, 16, 17, 18, 24, 32, 34, 40, 48, 50, 56, 64, 65, 72, 80, 81, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, … Continue reading →

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The geometry of diagonal groups

Posted on 18/04/2020 by

This is an interim report on ongoing work with Rosemary Bailey, Cheryl Praeger and Csaba Schneider. We have reached a point where we have a nice theorem, even though there is still a lot more to do before the project … Continue reading →

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More on derangements

Posted on 04/04/2020 by

Francis Bacon, in The New Organon, developed a famous metaphor: Those who have handled sciences have been either men of experiment or men of dogmas. The men of experiment are like the ant, they only collect and use; the reasoners … Continue reading →

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