Number of homomorphism between $\mathbb{Z}_3$ to itself

Question

How many homomorphisms exist from $\mathbb{Z}_n$ to itself? In particular I want to know how to find the homomorphisms between $\mathbb{Z}_3$ to itself.

Answer 1

Any homomorphism $\phi : \mathbb{Z}_3 \rightarrow \mathbb{Z}_3$ is completely determined by $\phi(1)$. So assume $\phi(1)$= $a$. Then $\phi(x)$= $ax$. Note that $o(\phi(1))$ divides 3. That is o($a$) divides 3. So o($a$) = 1 or 3. Thus $a$ = 0,1,or 2. So there are three homomorphisms namely $x$ $\mapsto$ $0$ ,$x$ $\mapsto$ $x$, $x$ $\mapsto$ $2x$.

In general, any homomorphism $\phi :\mathbb{Z}_n \rightarrow \mathbb{Z}_n$ is of the form $\phi(x)$= $ax$ ( one can easily check this defines a homomorphism) where $a$ $\in$ $\mathbb{Z}_n$. So there are totally $n$ candidates for $\phi$.