Order of Automorphism Group of Cyclic Group

Theorem
Let $C_n$ denote the cyclic group of order $n$.

Let $\Aut {C_n}$ denote the automorphism group of $C_n$.

Then:
 * $\order {\Aut {C_n} } = \map \phi n$

where:
 * $\order {\, \cdot \,}$ denotes the order of a group
 * $\map \phi n$ denotes the Euler $\phi$ function.

Proof
Let $g$ be a generator of $C_n$.

Let $\varphi$ be an automorphism on $C_n$.

By Homomorphic Image of Cyclic Group is Cyclic Group, $\map \varphi g$ is a generator of $C_n$.

By Homomorphism of Generated Group, $\varphi$ is uniquely determined by $\map \varphi g$.

By Finite Cyclic Group has Euler Phi Generators, there are $\map \phi n$ possible values for $\map \varphi g$.

Therefore there are $\map \phi n$ automorphisms on $C_n$:


 * $\order {\Aut {C_n} } = \map \phi n$