Automorphisms of a cyclic group

Proposition. Let $G=\langle g\rangle$ be a cyclic group of finite order $n$ . The map sending an automorphism $\alpha$ to the unique residue class $k$ satisfying $\alpha(g)=g^k$ gives an isomorphism

\mathrm{Aut}(G)\ \cong\ (\mathbb Z/n\mathbb Z)^\times,

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

Equivalent characterizations

The powers $g^k$ that generate $G$ are exactly those with $\gcd(k,n)=1$ , and composition of automorphisms corresponds to multiplication of their exponents modulo $n$ .

Remarks

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.