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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 $(S N) / N \cong S /(S \cap N)$.

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    $\begingroup$ The two statements are both true, so it doesn't really mean much to say that you can (or cannot) infer one from the other. Given any two theorems $A$ and $B$, there may or may not be "nice" ways of proving $A$ using $B$ as a lemma or vice versa. $\endgroup$ Commented Jul 17, 2020 at 21:53
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    $\begingroup$ In complete generality, the only options for $S$ are $1$, $N$ and $G$, and none of those works. So I don't think you can go the other direction, no. $\endgroup$ Commented Jul 17, 2020 at 22:13
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    $\begingroup$ @MorganRodgers Could you please read the titles slowly? They are complerely different questions. If you agree with me, please remove your downvote. $\endgroup$ Commented Jul 18, 2020 at 5:51
  • $\begingroup$ 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). $\endgroup$ Commented Jul 18, 2020 at 13:56
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    $\begingroup$ Hi @halrankard That's exactly what I was thinking in writing down this question. $\endgroup$ Commented Jul 18, 2020 at 14:12

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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).

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