= Automorphism of a finite cyclic group
{title2=$\operatorname{Aut}(C_n)\cong(\mathbb Z/n\mathbb Z)^\times$}
= Automorphisms of a finite cyclic group
{synonym}
Write the <finite cyclic group> as $C_n=\langle r\rangle$. 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 $\operatorname{Aut}(C_n)\cong(\mathbb Z/n\mathbb Z)^\times$. Its <group order> is therefore <Euler's totient function> $\phi(n)$. A <group homomorphism> $C_2\to\operatorname{Aut}(C_n)$ is determined by a <unit> $a$ with $a^2\equiv1\pmod n$.
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