Automorphisms of a cyclic group

Proposition (Automorphism group of a finite cyclic group). Let $G$ be a cyclic group of order $n$, and identify $G\cong \mathbb Z/n\mathbb Z$ via the standard isomorphism . Then

where $(\mathbb Z/n\mathbb Z)^\times$ denotes the group of units of the ring $\mathbb Z/n\mathbb Z$.

Equivalently: if $G=\langle g\rangle$, then every automorphism $\alpha\in \mathrm{Aut}(G)$ is uniquely determined by $\alpha(g)=g^k$ with $\gcd(k,n)=1$, and composition corresponds to multiplication of $k$ modulo $n$.

Context. This makes automorphisms of cyclic groups completely explicit: an automorphism is exactly the choice of a generator-image. The group $\mathrm{Aut}(G)$ itself is a central object in extension theory and semidirect products.