Equivalent Definitions of Normal Subgroup | Abstract Algebra

We prove a subgroup is normal (closed with respect to conjugates) if and only if its left and right cosets are equal. Thus these are equivalent definitions of normal subgroup. Some texts define normal subgroups as being closed with respect to conjugates, others define normal subgroups as having equal left and right cosets, but these characterizations are equivalent. #abstractalgebra

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[WrathofMath]

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[Bedoroski]

Nice layout that makes putting in effort worth it

[tomkerruish]

Personally, I prefer the statement that a normal subgroup is the kernel of a group homomorphism, as it tells me why I should even care. In particular, simple groups only have 'uninteresting' homomorphisms, with their images either being isomorphic copies or the identity element. Likewise, ideals and rings.

[golden_smaug]

I hadn't been able to understand why one definition implied the other, although I think I prefer the conjugate closure one. Thanks for the video :)

[voyager]

I have a question here...

In the first part of proof... While assuming the definition and proving the left coset equal to right coset

You showed that left coset is a subset of right coset ... Bcz aha-¹ belongs to H

But when you are proving that right coset is equal to left coset... Then you have used that (a-¹ • h • a) belongs to H. But rhats not conjugate right? I mean it is... But it is the other way around ... Instead of aha-¹ it it a-¹ha?
Does that hold for normal subgroup...

[MrCoreyTexas]

LOL, the further you get into the playlist the fewer the comments as the abstraction becomes more abstract. I feel like i'm in uncharted territory here. So do you call non normal subgroups abnormal or weird ?

[kabirbhattacharyya]

One thing that I'm not getting is in the condition aH=Ha, is it talking about every element in H? Well it should, but then again in my books it says aH=Ha doesn't demand for every h. I'm confused.