While it's quite standard to infer the second isomorphism theorem from the first one, I've not seen the proof in reverse direction. Is it possible to infer the first isomorphism theorem from the second one? Thank you so much for your help!
I tried to let $S := G$ and $N := \ker(\varphi)$, but this leads to a trivial equality.
Here is first isomorphism theorem:
Let $\varphi: G \to H$ be a group homomorphism. Then $G/\ker(\varphi) \cong \operatorname{Im}(\varphi)$.
Here is second isomorphism theorem:
Let $G$ be a group, $S \le G$, and $N \trianglelefteq G$. Then $(S N) / N \cong S /(S \cap N)$.
To follow up on the comment of @RobArthan, here's part of the trouble.
What does it mean to "infer the first isomorphism theorem from the second"?
Perhaps it means that one accepts the second isomorphism theorem as an additional axiom of group theory, and in light of that one writes down a proof of the first isomorphism theorem. The trouble is, I could just ignore the light altogether and write down any proof of the first isomorphism theorem (in fact, I can even write a very short proof of the first isomorphism theorem, which ignores the light).