Basic abstract algebra, pt.8

This is a video series about basic abstract algebra and group theory. We will keep it basic so that anyone can follow the videoes.

This is the eighth video in the series. We define abelian groups and subgroups.

Category:

Education

Tags:

• basic
• abstract
• algebra
• group
• theory
• mathematics
• math
• theorem
• prove
• proof
• science
• axioms
• subgroup
• abel
• abelian

[fstenv]

As I understand from this series that group is something like class or object from OOP languages such as C++, Java and etc. Did I understand correctly?

[VeritySeeker]

@fstenv Hi, no no really. Groups are way more general than that. Furthermore; groups have clear definitions. What the objects contain are not really relevant. All they need is to have some way of combining them in a binary sense :).

[eedahl]

Please tell me if I've gotten this right: if (S, *) and (S', *') are two groups S and S', they are _isomorphic_ iff there exists a bijective homomorphism F between the two. Where _homomorphism_ is simply defined by F(a * b)=F(a) *' F(b), and the isomorphism is the bijective part? Eg., a homomorphisms is just a tool for proving an isomorphism?

[VeritySeeker]

Two groups S and S' are isomorphic if and only if there exist a bijective homomorphism between the groups - that is correct.

Your definition of homomorphism is correct.

An isomorphism is a bijective homomorphism, by definition.

A homomorphism is not only a tool for defining isomorphism. Homomorphisms are important on their own. If there exist a homomorphism from S to S', then S and S' are homomorphic - which also gives information on its own.

[eedahl]

I meant "finding an isomorphism", not defining, but yeah, now I get it. Thanks a bunch :-)

[VeritySeeker]

My pleasure :).

[magestaff567]

Thanks so much for these videos. You have helped so many people!

[Semidicht]

First I thought, that the music would be distrubing and that it would be best to turn it off, but No Way! It's just perfect. It establishes an emotional connection and in such way it makes you more attentive and you can grasp more by inuition. or something like that (you should check the applied biology guys on that one)

[pinkypie10157]

..........what? O.o

[frozenphoenixprod]

@seniorgomez666 I don't remember the name but it's by a group called Infected Mushroom.

[seniorgomez666]

interesting. but also, what is the name of this song?!

[MinGophers]

Seriously. Thank you for making these videos.

[tharinduuuu]

nyyyyyyyyyyyyyyyyce image there at the end

[giovanni9107]

I loved the comic strip at the end!

[chrisdesilva66]

I understood the difference between isomorphism and homomorphism, and most of all the example of two isomorphic groups being Abelian was great - many thanks!!!