kernel

Let $\rho :G\to K$ be a group homomorphism. The preimage of thecodomain identity element $e_K\in K$ forms a subgroup of the domain$G$, called the kernel of the homomorphism;

$$\ker (\rho )=\{s\in G\mid \rho (s)=e_K\}$$

The kernel is a normal subgroup. It is the trivial subgroup if andonly if $\rho$ is a monomorphism.