After classifying cyclic groups, we can describe homomorphisms from cyclic groups with remarkable efficiency. The focus keyword homomorphisms between cyclic groups means that once the image of a generator is known, the whole homomorphism is known. If $G=\langle a\rangle$, every element of $G$ is a power of $a$, so a homomorphism must satisfy $f(a^k)=f(a)^k$. The only restriction is that the chosen image of $a$ must respect the order of $a$. This lesson turns the abstract homomorphism condition into a practical listing method for maps between finite cyclic groups and from $\mathbb{Z}$ to cyclic groups.

Let $G=\langle a\rangle$ be a cyclic group and let $f:G\to H$ be a homomorphism. Then $f$ is completely determined by $f(a)$. If $o(a)=n<\infty$, then $$\begin{equation*} o(f(a))\mid n. \end{equation*}$$