# Counterexample: Two groups $H$ and $G$, with surjective homomorphismus

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.

• Hint: take a non-abelian simple group $H$, and take $G=N$ any proper subgroup of $H$ May 28 '20 at 19:55

Every group homomorphism $$\Bbb Z\to S_3$$ which maps $$1$$ into an odd permutation.
• Thank you! Should I consider $\mathbb{Z}$ with Multiplication or Addition? May 28 '20 at 20:05