Category Archives: exposition

a post aimed to teach something

Graphs on groups, 5

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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Oligomorphic groups: topology or geometry?

One perhaps unexpected result of the pandemic is that there is a huge volume of really interesting mathematics flying around the internet at the moment, courtesy of Zoom and other platforms. This week I went to a talk by Joy … Continue reading

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Graphs defined on groups

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

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

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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The Fitting subgroup

I have talked a bit about the Frattini subgroup. Time for its big brother. The definition of the Fitting subgroup F(G) of a finite group G is the unique maximal normal nilpotent subgroup of G. As such, of course, it … Continue reading

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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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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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Puzzle solution

Thank you, Honza, spot on. In 1964, Richard Rado published a construction of a universal graph, a countable graph which embeds every finite or countable graph as an induced subgraph. His graph turns out to be an explicit example of … Continue reading

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Peter Sarnak’s Hardy Lecture

Yesterday, Peter Sarnak gave the London Mathematical Society’s 2020 Hardy Lecture (remotely). He talked about gaps in the spectra of connected cubic graphs. It was a talk properly described as a tour de force, applying to the problem ideas from … Continue reading

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