Kernel and First Isomorphism Theorem - Group Theory
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- čas přidán 27. 07. 2024
- 0:00 Kernel is a Normal Subgroup
5:20 First Isomorphism Theorem
The first isomorphism theorem is a fundamental theorem in group theory that gives us a powerful way to find isomorphic groups. In this video, we explain what the kernel of a homomorphism is and how to turn a homomorphism into an isomorphism!
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Music: C418 - Pr Department
I love how you always give a quick review on the process at the end 😊
This method of quotienting by the kernel in order to fix injectivity is much more enlightening than the normal proof of defining psi first, and just showing that it is well defined and an isomorphism. Thanks!
Very, very, very well explained ☺. I wish, i could have seen this video 5 years ago.
Simply wonderful. A way of proof that is not seen elsewhere and is natural, Bravo.
Nice! I think this video gave me a insight: the procedure to find the isomorphism is like when we have a function and we want a bijection out of it. The case of the function say F:A -> B, we can simply restrict it to its image, to turn it surjective, and group together the elements of the inverse image of each element of Im F, to turn it injective. This give us a bijection F: Disjunct union of the inverse image of each element of Im F -> Im F. What happens in group theory seems to be just a abstraction of this procedure, and the relations we have on the kernel give us a more natural way to group together the elements of the domain of the function. And I guess this procedure is done for other algebraic structures the same way.
Thanks for this, helps so much!
This was so incredibly helpful, thank you so much!
This is very clear and helpful👍
Great video!
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