Intuition behind the third isomorphism theorem.

Question

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!

Answer 1

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"