AlgTopReview4: Free abelian groups and non-commutative groups

Free abelian groups play an important role in algebraic topology. These are groups modelled on the additive group of integers Z, and their theory is analogous to the theory of vector spaces. We state the Fundamental Theorem of Finitely Generated Commutative Groups, which says that any such group is a direct sum of a finite number of Z (integers) together with a finite commutative group (the torsions part).

Non-commutative groups are also briefly introduced, mostly through the simplest example of S_3, the symmetric group on three objects, or alternatively the group of 3x3 permutation matrices. We illustrate that cosets of a subgroup in the commutative case are of two different types: left and right, and this makes the situation more complicated. When the left and right cosets agree we are in the situation of a normal group, and then the cosets do form a quotient group.

Thanks to Nguyen Le for filming.

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[rainks]

That professor he understand deeply Algebra. I love him.

[njwildberger]

Both are nice topics that I would like, in the fullness of time, to do some more videos on. Thanks for the suggestion, and for all that watching!

[MistressGlowWorm]

Great lecture series. I love to see Abstract Algebra and Topology!!

[jenhjjang]

Amazing video! Thanks for filming such a nice lecture!

[harrisonbennett]

Fantastic Video, your an amazing explainer thank you!

[njwildberger]

Yes that is right, but here we are working over a field, not a general ring, which is a much more rigid and predictable situation.

[gibsonrulezz]

Brilliant videos. Would it be easy for you to make some more algebra videos (about rings, fields etc) ? Maybe some Galois Theory would be nice too.

[jeffreyhowarth]

Thanks for such a mesmerizing example around 20 minutes. Is that an example of a group without a basis?

[LearningAutomata]

...unsolved computational problems can be solved within it. Of course the challenge will be in seeing the resemblance and doing the proper transformation

[njwildberger]

It's a free abelian shirt. And thanks.