Image of the identity under a group homomorphism is the identity

For any Group homomorphism $f: G \to H$, we have $f(e_G) = e_H$ where $e_G$ is the identity of $G$ and $e_H$ the identity of $H$.

Indeed, $f(e_G) f(e_G) = f(e_G e_G) = f(e_G)$, so premultiplying by $f(e_G)^{-1}$ we obtain $f(e_G) = e_H$.