Automorphism of a finite cyclic group ($\mathrm{Aut}(C_n)\cong (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$)

Write the finite cyclic group as $C_n=⟨r⟩$. A group homomorphism is determined by $r\mapsto r^a$; it is a group automorphism exactly when $r^a$ generates $C_n$, equivalently $\gcd (a,n)=1$. Composition multiplies the exponents modulo $n$, proving $\mathrm{Aut}(C_n)\cong (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$. Its group order is therefore Euler's totient function $\varphi (n)$. A group homomorphism $C_2\to \mathrm{Aut}(C_n)$ is determined by a unit $a$ with $a^2\equiv 1(\mathrm{mod}n)$.