In group theory we have the second isomorphism theorem which can be stated as follows:
Let $G$ be a group and let $S$ be a subgroup of $G$ and $N$ a normal subgroup of $G$, then:
- The product $SN$ is a subgroup of $G$.
- The intersection $S\cap N$ is a normal subgroup of $G$.
- The quotient groups $SN/N$ and $S/(S\cap N)$ are isomorphic.
I've seen this theorem some time now and I still can't grasp an intuition for it. I mean, it certainly is one important result, because as I've seen it is highlighted as one of the three isomorphism theorems.
The first isomorphism theorem has a much more direct intuition though. We have groups $G$ and $H$ and a homomorphism $f:G\to H$. If this $f$ is not injective we can quotient out what is stopping it from being injective and lift it to $G/\ker f$ as one isomorphism onto its image.
Is there some nice interpretation like that for the second isormorphism theorem? How should we really understand this theorem?