Chapter 6: Homomorphism and (first) isomorphism theorem | Essence of Group Theory

The isomorphism theorem is a very useful theorem when it comes to proving novel relationships in group theory, as well as proving something is a normal subgroup. But not many people can understand it intuitively and remember it just as a kind of algebraic coincidence. This video is about the intuition behind the idea of homomorphism - functions preserving group structures, and the closely related isomorphism theorem.

There are second, third and even fourth isomorphism theorem (the fourth one is usually disputed), but all can be derived from the first one, using clever constructions of homomorphisms.

Apparently, when I was typing the description (after the video is edited), I knew that the name "homomorphism" is probably mistranslated from German. Originally, it was supposed to mean "similar", not "same".

This video series is about understanding the group theory intuitively, complementing how most people learn about it, because it is usually introduced as part of abstract algebra.

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[mathemaniac]

Like and share this video series if you think this video series is useful or just enjoy these videos in general. Also, don't forget to subscribe with notifications on!

[MikhailBarabanovA]

Bored you said? These lectures are diamonds!

[abjectindividual]

Putting the mathematical rigor/jargon click into place with such enlightening expositions is so satiating, thank you.

[bobtheblob]

this series is so good!! I'm so glad I stumbled upon it. it looks like the YouTube algorithm is starting to recommend to more people, I hope that translates to more people appreciating your fantastic content :)

[Alpasonic]

Talking about technical aspects of your videos - I think your way of of the words and the speed of your talk is just optimal. And perfectly relevant to the Purpose. Thank you for your videos. And best wishes.

[abbasmehdi]

Sad to see so less views....i know that u will lead...keep up the great content.

[kabirbelgikar]

I'm currently not at the stage where these (group theory) videos will help me very much (just started discrete math very recently). However, knowing the quality of your videos I'm sure that they'll help me a lot when I decide to learn this beautiful subject. Don't ever stop making these! I'm sure your channel will blow up soon.

[marcocecchi]

Keep this going!
Loving this series, if you keep posting more i'll be sure to follow

[pepepepe]

Thanks for doing this series! Loved it. I've been self studying group theory (which means I'm learning proofs of the theorems along with intuitions) and these videos were helpful. I often find that proofs are not difficult to do when you have a solid intuition of the concept.

[sebastianmarshall]

Honestly this content is unreal. Thank you for helping me with Math 113!

[escher]

It 'd be interesting if you show an application of group theory where it's really essential for the proof. Some problem that could not or hardly be solved in a strait forward way

[PunmasterSTP]

Homomorphism? More like "Hurry up, this is incredible, isn't it?" Your videos are so good, and I can't wait to watch the rest!

[rajaryan]

But what should e the value of phi to prove the second isomorphism from the first one? And how do we do it? It would be great if someone could throw some light on this, any hit would work, PLEASE.

[hyperduality]

Same is dual to different.
The normal subgroup is dual to homomorphism (factor group) synthesizes the kernel.
The image (co-domain) is a copy, equivalent or dual to the factor group (domain) - the 1st isomorphism theorem.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference).
Injective is dual to surjective synthesizes bijective or isomorphism.
Similarity, equivalence = duality!
Isomorphism represents the orthogonal complement or the dual of the kernel.
Homo is dual to hetero.

[johnhippisley]

doesn’t an isomorphism also have to be onto?

[FunctionalIntegral]

Is there any reason that you permanently emphasis words you are saying? I mean you don't need to give yourself too much pressure.
--> Subgroup
-> Rotation
Loosen up a little!

[tricanico]

Thank you!!
It makes me a little bit sad how many views this has. But people will realize sooner or later!

[joaofrancisco]

I just started watching this series now, but I just came here to say that I'm not bored at all!!! Your videos are great :)

[vishwanathlohar]

Plz keep uploading more videos..
Good explainaton.

[abbasmehdi]

Well ! I do not understand the video but i know that it will be intresting and informative for me when i will move in higher classes. Never stop uploading such videos . Earlier I thought symmetries are not intresting but your content changed my ideas.