# Tag Archives: symmetric group

## A theorem on polytopes

You know what polygons and polyhedra are. How do we extend their study to higher dimensions? There are two parts to this question. The first involves incidence geometry: vertices, edges, faces, etc. Here the generalisation is fairly straightforward. A polygon … Continue reading

Posted in doing mathematics, exposition | | 3 Comments

## June

This month, a beautiful formula on an old door panel in Prague. (It may not be very legible in this low-resolution copy.) The formula is for l(Sn), the length of the longest chain of subgroups in the symmetric group Sn. … Continue reading

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## 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

## Regular polytopes, 3

In the last two posts on regular polytopes, I gave away something about my method of working. Although I have known about regular polytopes for a long time, I have never attempted to do research on them before. I find … Continue reading

## From the archive, 4

A photocopy of a sheet of paper in my handwriting. At the top left, my initials are written in the handwriting of Jaap Seidel. The page begins as follows. Theorem. Let X be a set of points in the n-cube … Continue reading

## The symmetric group, 13

In my post on the Novi Sad Algebraic Conference, I described how the symmetric group on a set X is “contained” in the full transformation semigroup on X, which is itself contained in the clone of functions on X. In … 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

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## 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