TITLE: Image of a normal subgroup under a homomorphism
QUESTION [5 upvotes]: I want to find an example of a group homomorphism $f:G\to H$ such that $A$ is a normal subgroup of $G$, but $f(A)$ is not so in $H$. 

Definitely the groups must be noncommutative and the function should not be onto. Please help me.

REPLY [6 votes]: For instance, you could take any non-normal subgroup $N$ of a group $G$, and let $f$ be the inclusion of $N$ into $G$. Obviously $N$ is normal in $N$, but it isn't normal in $G$...
Can you find an example where $N$ is commutative?