# Category Archives: symmetric group

one of my favourite mathematical objects

## 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

## A family of non-synchronizing groups

As I explained recently, according to the O’Nan–Scott Theorem, a finite primitive permutation group either preserves a Cartesian structure, or is of affine, diagonal or almost simple type. In all these types except the last, the action of the group … Continue reading

## Chains of semigroups

I have written here about the lovely formula for the length of the longest chain of subgroups in the symmetric group Sn: take n, increase it by 50% (rounding up if necessary), subtract the number of ones in the base … Continue reading

## Remoteness

After the last bit of bureaucratic nonsense, what a relief to turn to mathematics again. Maximilien Gadouleau and I have just submitted a paper about a concept for finite metric spaces somewhat related to domination, which we call remoteness. It … Continue reading

## The symmetric group, 12

This instalment is about the maximal subgroups of the symmetric group, and the O’Nan–Scott Theorem. There are two versions of this theorem, one of which is sometimes called the Aschbacher–O’Nan–Scott Theorem. One is about maximal subgroups of the symmetric group … Continue reading

Posted in exposition, symmetric group | | 6 Comments

## The symmetric group, 11

I am going to talk about a celebrated theorem of John Dixon and some of its variants; this is on my mind at the moment, for reasons I will explain at the end. Dixon’s theorem is easily stated. Two random … Continue reading

Posted in exposition, symmetric group, synchronization | | 4 Comments

## The symmetric group, 9

There are many different ways to regard a finite symmetric group as a metric space. These have been used for various purposes. Here I would like to say something about them. One application of such metrics is in non-parametric statistics. … Continue reading

Posted in exposition, mathematics, symmetric group | 2 Comments

## The symmetric group, 6

I quote here the review on MathSciNet for a paper which appeared in a conference proceedings in 1970. (This review, of course, was published in Mathematical Reviews, and now has been put on the website.) I have made one correction, … Continue reading

Posted in books, exposition, symmetric group | Tagged | 4 Comments

## The symmetric group, 5

Time to say a bit about the combinatorics of the symmetric group. Certainly not a complete survey of a vast topic, which would be quite impossible; I will touch on just a few things. Loosely speaking, the subject divides into … Continue reading

Posted in exposition, open problems, symmetric group | 4 Comments

## The symmetric group, 4

One of my views on teaching abstract algebra, with which my colleagues don’t all agree by any means, is that it is better to do rings before groups. The view for putting groups first is that they have just a … Continue reading

Posted in exposition, symmetric group | 7 Comments