Describe all group homomorphisms from $\mathbb Z$ to $D_8$.
So $\mathbb Z=\langle1\rangle$, and $f(1)$ determines the homomorphism. So every homomorphism is determined by $f(1)=\sigma$, and for some $m \in Z$, $f(m)=\sigma^m$. How can I find the number of homomorphisms from here though? What should I look at?
Thanks in advance!