Homomorphism and images

Question

G is a finite group, $\phi:G \to G$ a homomorphism. $\psi:G \to G$ is a homomorphism defined by $\psi(x)=\phi(\phi(x))$. Prove that $(\ker\phi= \ker \psi)\implies($Im$ \psi=$Im$ \phi)$. Can someone help me with this problem?

Answer 1

Since $\text{Im}\, \phi \subseteq G$, $\text{Im}\, \psi \subseteq \text{Im}\, \phi$. On the other hand, $$|\text{Im}\, \phi| = (G : \text{ker}(\phi)) = (G : \text{ker}(\psi)) = |\text{Im}\, \psi|.$$

Hence $\text{Im}\, \psi = \text{Im}\, \phi$.