Kernel and Image of a Group Homomorphism

Let $(G,\ast_G)$ and $(H,\ast_H)$ be groups, let $e_H$ be the identity element of $H$ as provided by Uniqueness of the Identity Element and of Inverses in a Group, and let $\varphi:G\to H$ be a group homomorphism.

The kernel of $\varphi$ is the subset of $G$ given by

$$\ker\varphi=\{a\in G:\ \varphi(a)=e_H\},$$

and the image of $\varphi$ is the subset of $H$ given by

$$\operatorname{im}\varphi=\{y\in H:\ y=\varphi(a)\ \text{for some}\ a\in G\}.$$