Category Archives: doing mathematics

how is it done?

On the Frattini subgroup

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

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

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

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

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

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

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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Au revoir, GRA

Today (Wednesday 18 March) the Groups, Representations and Applicatons programme at the Isaac Newton Institute came to a premature end. There are still some hopes that it can be revived later, but at the moment the only certainty is that … Continue reading

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Research trips in February

I don’t know how things have got so busy. I had two interesting trips in February; I worked hard, and some interesting mathematics resulted; but I don’t seem to have found the time to describe it. So here goes. This … Continue reading

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Aliens Do Exist

The people from the planet Ade have intercepted radio transmissions from Earth, and have discovered that we know about the Petersen graph and the root system E6. One day, a flying saucer from Ade arrives on Earth and delivers an … Continue reading

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