A Natural Proof of the First Isomorphism Theorem (Group Theory)

The first isomorphism theorem is one of the most important theorems in group theory, but the standard proof may seem artificial, like every step of the proof is set up knowing that we're trying to create an isomorphism. In this video, we show an alternate proof with no such tricks using the preimage map of a group homomorphism.

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0:00 Setup
6:18 Homomorphism
8:50 Injective
10:35 Surjective

Music: C418 - Pr Department

[lexinwonderland]

So glad to see you back posting videos again! I think you always explain things in a very clear and direct way (without being too impersonal or lecture-like), and I especially appreciate when you cover more complex topics like group theory/abstract algebra, tensors, higher level differential equations, etc. because it's so difficult to find resources about otherwise. I look forward to the next one! Have a great day! :)

[lenoel]

Very nice explantion and indeed a very natural approach! Thank you very much.

[tetianasokolova]

Thank you for a very beautiful and clear explanation! I already know more formal way to prove the first isomorphism theorem, which you mentioned in the beginning. However, I always felt that I lack a more natural approach and some intuition behind this theorem. After watching this video I can finally understand it :) Thank you!

[haiderzia]

wonderful lecture with grear explanation

[raulbeienheimer]

Smart and handsome, thanks for the quality content!.

[fuyikuang]

Thank you for making this video! So helpful.

[MuffinsAPlenty]

I like this perspective on the first isomorphism theorem, so thanks for sharing!

I wouldn't describe the typical proof as using "tricks" though. If one develops a good intuition for what quotient groups mean and also a good intuition for how the kernel relates to injectivity, the typical proof of "making ϕ injective" is pretty straightforward.

G/N should be thought of as carefully collapsing G in a way which makes every element of N collapse to the identity. Making N collapse to the identity then forces other elements to collapse to each other. For example, if g = hn for some element n in N, then since n is becoming the identity in this "collapse, " after collapsing, we must have g = he = h. So g must collapse to h as well. And similarly, if g = nh, then after the collapse, we have g = eh = h, so g = h. This also explains exactly why we need left- and right cosets to be equal to each other, i.e., why we need a normal subgroup.

If one shows, as you showed, that two elements are mapped to the same output if and only if they differ by an element in the kernel, then one sees that, going modulo the kernel, two elements map to the same output if and only if they collapse to same element in the quotient G/kerϕ. So yes, going modulo the kernel "collapses" the non-injectivity of ϕ.

Now, it takes some time to develop good intuition for quotients. But what you call "tricks, " I really see as "the intuition one should have for quotients." And I think it's not a good idea to disparage good intuition as "tricks".

[alejrandom]

6.6k views and only 17 comments? Let me fix that

[justanotherman]

@5:50 how is the kernel of phi normal subgroup. Doesn't this fact use this fact to begin with?