Burnside's Lemma (Part 1) - combining group theory and combinatorics

Описание: A result often used in math competitions, Burnside's lemma is an interesting result in group theory that helps us count things with symmetries considered, e.g. in some situations, we don't want to count things that can be transformed into one another by rotation different, like in this case - how many ways are there to paint a cube's faces when we are given three colors, if two coloring patterns are considered the same when they differ just by a rotation?

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There is usually also something called Pólya Enumeration Theorem, which is a generalization of Burnside's lemma, and can be used to tackle a wider set of problems, but most problems that Pólya Enumeration Theorem can be applied to can also be tackled by Burnside's lemma, so this is usually more important. The theorem simply extracts the core of what we are doing and put it in a nice generating function, which can be useful, but the notation and the computations required are very troublesome, and does not fit well too well with the theme of this channel.

Non-mathematical applications like counting the number of isomers of an organic molecule (organic chemistry) and the number of trichords (music theory) are usually tackled by the theorem mentioned above, but this can really be tackled by Burnside's lemma, just with a bit more care. We will explore how this can be applied in those situations in the next video.

By the way, this lemma is not actually first discovered by Burnside, and the Pólya Enumeration theorem is also not first discovered by Pólya, but this phenomenon is also prevalent throughout mathematics and science, known as Stigler's law of eponymy.

This is not a part of the "Essence of Group Theory" video series, because it is not "essence", but an application of the orbit-stabilizer theorem, which is in Chapter 2 of the video series:    • Chapter 2: Orbit-Stabiliser Theorem | Esse...  

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