Homomorphic image of a nilpotent group is nilpotent Is it true that an image of a nilpotent group under a homomorphic function is nilpotent? In that case, how am I going to show this? Thanks in advance.

Do you know the correspondence theorem for subgroups? You can think of a homomorphic image of $G$ as a quotient $G/N$. If $G$ is nilpotent then the lower central series terminates. You just have to show that the lower central series for $G/N$ terminates.