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 $C_n$:

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

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64 \delta$