I'm trying to prove the following statement:
If $\phi :G\rightarrow H$ is a group homomorphism and $G$ is soluble, then $Im(\phi)$ is also soluble,
I tried creating a map $\psi:G\rightarrow Im(\phi)$, such that $\psi(g)=\phi(g)$, where $g\in G$. Clearly the map is surjective, but how can I show that it's injective. By doing so, I get that $G\cong Im(\phi)$. Would this be enough to show that $Im(\phi)$ is soluble.