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I was looking for intuition for the isomorphism theorems, and particularly for the third one. I read the post here but it didn't really help me.

I am looking more for an intuition like for the one given in his comment by David Wheeler in here.

If someone has such a short and clear explanation, I would really appreciate. Thank you!

ArsenBerk
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roi_saumon
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    Why didn't the two posts help you? They have a lot of nice answers and comments. If you like a short and clear explanation, then just repeat the comment there:"I always thought of it as a cancellation property". – Dietrich Burde Sep 21 '18 at 21:04
  • Here's a quote from Dummit & Foote: "[The third isomorphism theorem] shows that we gain no new structural information from taking quotients of a quotient group." – Henry May 18 '21 at 23:57

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Since the OP wanted something in the style of David Wheeler's comment, perhaps this answer (with shamelessly stolen verbiage) will be satisfying:

Suppose we have a subgroup of $G$ with two normal subgroups $K$ and $N$, such that $K\subseteq N$. We might want to know "what happens when we quotient out $N$ from $G/K$". The trouble is, elements in $G/K$ are cosets (of $K$ in $G$), and so $N$ is not even a subgroup of $G/K$. So what do we mean by this?

  • On one hand, part of the third isomorphism theorem states that $N/K$ (that is, the collection of cosets with representatives in $N$), is a normal subgroup of $G/K$, and so in that sense we might choose to take the quotient $\dfrac{G/K}{N/K}$.
  • On the other hand, remember that $K\subseteq N$, and the quotient operation is supposed to "send the normal subgroup to the identity". So quotienting $G$ out by $N$ should send everything in $K$ to the identity anyway, so we might choose to simply take $G/N$.

The third isomorphism theorem states that both approaches lead to "the same place"

aleph_two
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