Symmetric Groups (Abstract Algebra)

Symmetric groups are some of the most essential types of finite groups. A symmetric group is the group of permutations on a set. The group of permutations on a set of n-elements is denoted S_n.

Symmetric groups capture the history of abstract algebra, provide a wide range of examples in group theory, are useful when writing software to study abstract algebra, and every finite group can be represented as a subgroup of a symmetric group. This result is known as Cayley's Theorem.

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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition

Milne, Algebra Course Notes (available free online)

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Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
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[AbhijitDas-wgmb]

My math examination is tomorrow and your videos will definitely help me pass this semester. I will recommend your videos to my friends as well. Although it's too late now.

[Socratica]

Hello everyone!  We just released our latest *Abstract Algebra* video.  In this lesson, we introduce the *Symmetric Group*.  We *define* it, give some *examples*, and introduce a *compact notation for permutations*.

Next up in the Abstract Algebra playlist: *homomorphisms*, *isomorphisms* and *kernels*.

#LearnMore

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[nothingisimpossible.]

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[mkpmgpp]

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[JeffreyAborot]

I have not taken Abstract Algebra or specifically Group Theory before or did not know I was using the concepts in it in some of what I am currently doing. After watching the series of videos I felt like if it is fun to learn Abstract Algebra and thought of the so many things one could explore in and with it.

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[bonbonpony]

Another useful application of permutation groups is in cryptography, because cryptography is all about rearranging the sets of symbols (e.g. letters of the alphabet) to make a different alphabet.

[twilightsparkle]

My favorite way of representing permutations is to arrange all the elements of a set in a circle/ring, and for every element (an input) draw an arrow to the element it maps to (its output). This is the most visual way of doing it, and it often help in revealing some patterns in the permutation.

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[sherazahmed]

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[pittdancer]

This makes so much sense connecting S3 back to the triangles!

[davidpal]

Very effective and simple way . Thank you so much . I was totally confused in this topic but after watching this video everything is clear to me about the permutation of Sn .

[markmathman]

At 0:12, S sub n equals the group of permutations on the set { 1, 2, 3, ..., n}. If A is a set of some other n elements, that symmetric groups is called S sub A.

[ugaleadinath]

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[avinashbhatia]

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