Automorphisms of a finite cyclic group

#group_theory

Let $Z_{n}$ denote the cyclic group of order $n$ . Suppose $m \in N$ is relatively prime to $n$ . Define the function $μ_{m} : Z_{n} \to Z_{n}$ by $m [a]_{n} = [m a]_{n}$ .

Prove that the map $μ_{m}$ is a well-defined automorphism of $Z_{n}$ .
Prove that any automorphism of $Z_{n}$ has the form $μ_{m}$ for some $m$ .

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Let ${\bf Z}_n$ denote the cyclic group of order $n$. Suppose $m\in {\bf N}$ is relatively prime to $n$. Define the function $\mu_m:{\bf Z}_n\to {\bf Z}_n$ by $m[a]_n=[ma]_n$.
\begin{enumerate}[label=\alph*)]
	\item Prove that the map $\mu_m$ is a well-defined automorphism of ${\bf Z}_n$.
	\item Prove that any automorphism of ${\bf Z}_n$ has the form $\mu_m$ for some $m$.
\end{enumerate}