Group homomorphism/Z to Z/Example

Let ${}d\in \mathbb {N}$ be fixed. The mapping

$\mathbb {Z} \longrightarrow \mathbb {Z} ,n\longmapsto dn,$

is a group homomorphism. This follows immediately from the distributive law. For ${}d\geq 1$, this mapping is injective, and the image is the subgroup ${}\mathbb {Z} d\subseteq \mathbb {Z}$. For ${}d=0$, we have the zero mapping. For ${}d=1$, the mapping is the identity. For ${}d\geq 2$, the mapping is not surjective.