How could I prove that Aut$(C_n) = (\mathbb{Z}/n)^*$

Question

Given that Aut$(C_n)$ is the set of automorphisms of the cyclic group of size $n$, and that $(\mathbb{Z}/n)^*$ is the multiplicative group of integers modulo $n$.

Answer 1

Define a family of maps, $\phi_s \colon C_n \to C_n$ by $x \mapsto x^s$. You must prove the following:

1. $\phi_s$ is an automorphism if and only if $s$ is coprime to $n$.

2. Every automorphism of $C_n$ is of the form $\phi_s$ for some $s$.

Do you see why both of these are true? Do you see why these 2 statements prove your result? If not I can add more hints.

Answer 2

If $\;C_n=\langle\,c\,\rangle\;$ , then it'd help to know that

(1) Any automorphism of a cyclic group sends a generator to a generator, and

(2) $\;c^k\;$ is a generator of $\;C_n\;$ iff $\;\gcd(n,k)=1\;$