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# Category Archives: exposition

## 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

Posted in doing mathematics, exposition
Tagged Anti-foundation Axiom, Bea Adam-Day, random graph
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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

Posted in doing mathematics, exposition
Tagged commuting graph, Lovász, partial preorder, perfect graph, power graph
1 Comment

## 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

Posted in exposition
Tagged Fitting subgroup, Frattini argument, nilpotence, Sylow's theorem
3 Comments

## 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

Posted in doing mathematics, exposition
Tagged Frattini subgroup, G. A. Miller, writing mathematics
4 Comments

## 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

Posted in doing mathematics, exposition
Tagged Carlo Casolo, derived subgroup, Sofos Efthymios
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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

Posted in exposition
Tagged countable random graph, Henson graphs, hereditarily finite set theory, Rado graph
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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

Posted in events, exposition
Tagged Alan Hoffman, fullerenes, generalised line graphs, Ramanujan graphs, spectral gaphs, waveguides
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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

Posted in doing mathematics, exposition
Tagged Cartesian lattices, diagonal groups, partitions
8 Comments

## The B. B. Newman Spelling Theorem

This is a guest post by Carl-Fredrik Nyberg Brodda, a recent Masters student at St Andrews and currently a PhD student at the University of East Anglia. The story has personal resonance for me, because it turns out that B. … Continue reading