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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
Tagged Dimitri Leemans, Maria Elisa Fernandes, regular polytopes, symmetric group, zoom
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
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
Posted in exposition, history
Tagged cube, Jaap Seidel, Johnson scheme, symmetric group
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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
Posted in exposition, symmetric group
Tagged maximal subgroups, O'Nan-Scott theorem, permutation group, 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
Tagged Baire category, Dixon's theorem, symmetric group
4 Comments
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