Let $\varphi:G\to G'$ be a group homomorphism. Is $\varphi(G)$ a normal subgroup of $G'$?
I dont know how to prove that the statement is false.
Without lose of generality suppose that $G$ is a subgroup of $G'$ and $\varphi$ is the inclusion map, then the problem is reduced to prove that not any subgroup is normal.
Then, how I can prove that not any subgroup is normal? I dont know any example or how I can derive this result from the axioms of groups. Can you help me please? Thank you.