Symmetric and Alternating Groups -- Part 1

In this video, we show how to express elements of the symmetric group S_n as products of 2-cycles (i.e., transpositions), and this allows us to define even permutations and odd permutations. From this, we are able to describe an important subgroup of S_n consisting of the even permutations: the alternating group A_n.

[jobjoseph]

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[deep]

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Therefore, you teach well and it is natural.
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[akhilaani]

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[danieljohnson]

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[martin]

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[divyaraghavan]

Simplest way to understand about symmetric group and alternating group. Thank you 🙏

[winstonjiang]

Thanks for sharing! The video really helps to visualize and recap everything when one comes out of abstract algebra class.

[TravellingDon]

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[esak]

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[onattanriover]

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[rikia.h]

Every time you do your little nervous laugh like at 9:58 it
makes me feel exposed for not getting it haha

[mathwithumair]

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[paulhammond]

It always seemed to me that you should just take a transposition (ab) as the easiest example of how the parity thing works - a transposition is written as a product of 1 transposition, one is an odd number, so a transposition is an odd permutation. Then (abc) = (ac)(ab) therefore 3-cycles are even. The technique of breaking up a cycle into transpositions already implies that the number of transpositions you will get will be equal to cycle length minus one.