Lemma: Kernel and Image of Group Homomorphism

Let $(G,\ast)$ and $(H,\cdot)$ be two groups with the respective identities $e_G$ and $e_H$ and $f:G\rightarrow H$ be a group homomorphism. We define:

the kernel $\operatorname{ker}(f):= \{g\in G\mid f(g)=e_H\}$, in other words, it is the fiber of $e_H$ under the group homomorphism $f$ ,
the image $\operatorname{im}(f):=f[G]=\{f(g)\in H\mid g\in G\},$ in other words, it is the image of $G$ under the group homomorphism $f$.

The kernel and the image of $f$ fulfill the following defining properties: