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# Category Archives: exposition

## 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

## 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

## Hall’s Marriage Theorem

Philip Hall was one of the greatest group theorists of the twentieth century. But it may well be that he is known to more people for a result which on the face of it is pure combinatorics, with nothing to … Continue reading

Posted in exposition
Tagged left cosets, Marshall Hall Jr., Philip Hall, right cosets, system of distinct representatives
13 Comments

## 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

## GRA workshop 2

This workshop was on “Computational and algorithmic aspects” (of groups, representations and applications, presumably). In my opinion, this was the week when the programme really took off. There were many good talks, so as ever I shall just select a … Continue reading

## GRA Workshop 1

Last week was the introductory/instructional workshop for the Isaac Newton Institute’s six-month programme on Groups, Representations and Applications. We were thrown in the deep end right at the start. The first two talks were on what were claimed to be … Continue reading

## The Frattini argument

The Frattini subgroup of a finite group G can be defined in two equivalent ways: it is the intersection of all the maximal proper subgroups of G; it is the set of all non-generators of G, that is, elements which … Continue reading

## Sylow’s Theorem, from the book

The most important theorems of elementary group theory are those of Lagrange and Sylow. I want to describe here what I consider the most beautiful proof of the first part of Sylow’s Theorem, actually based on Sylow’s original proof. But … Continue reading

## Something I didn’t know

I didn’t know this, though probably I should have. Maybe you didn’t know it either. We work in a semigroup, a system with an operation (called multiplication) satisfying the associative law. A generalised inverse of an element A is an … Continue reading

Posted in exposition
Tagged associative law, commute, generalised inverse, matrices, polynomial
4 Comments