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So I need to find a counter-example, i.e. I need to find two groups $H$ and $G$, with $N$ being normal subgroup of $G$, with a NON-surjective Homomorphism $\phi: G \rightarrow H$, such that $\phi(N)$ is NOT normal subgroup of H.

I know for sure, that H isn´t supposed to be abelian group, but I am not sure which one to take! I would appreciate any kind of help.

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  • $\begingroup$ Hint: take a non-abelian simple group $H$, and take $G=N$ any proper subgroup of $H$ $\endgroup$
    – Exodd
    May 28 '20 at 19:55
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Every group homomorphism $\Bbb Z\to S_3$ which maps $1$ into an odd permutation.

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  • $\begingroup$ Thank you! Should I consider $\mathbb{Z}$ with Multiplication or Addition? $\endgroup$ May 28 '20 at 20:05
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    $\begingroup$ Addition; also a multiplicative infinite cyclic group works $\endgroup$ May 28 '20 at 20:06
  • $\begingroup$ shouldn't a group homomorphism map the identity into the identity? $\endgroup$
    – Exodd
    May 28 '20 at 20:08
  • $\begingroup$ You are right, I mean 1, fixed the typo, thank'you $\endgroup$ May 28 '20 at 20:10
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    $\begingroup$ Yeah, this is a good counter example! $\endgroup$ May 28 '20 at 20:16

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