TITLE: Homomorphism from $\mathbb{Z}/n\mathbb{Z}$
QUESTION [7 upvotes]: Does there exist a non-zero homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly?

REPLY [2 votes]: Let $\varphi : \mathbb{Z}_n \to  \mathbb{Z}$ be a homomorphism. Using isomorphism theorem, $\mathbb{Z}_n / \text{ker}(\varphi)$ is isomorphic to $\text{Im}(\varphi)$. Because $\text{Im}(\varphi) \leq\mathbb{Z}$, either $\text{Im}(\varphi)= \{0\}$ or $\text{Im}(\varphi) \simeq \mathbb{Z}$; on the other hand, $\mathbb{Z}_n/ \text{ker}(\varphi)$ is necessarily a finite group. Therefore, $\varphi =0$.