# 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.