# Order of Automorphism Group of Dihedral Group

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## Theorem

Let $D_n$ denote the dihedral group of order $n$.

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

Then:

- $\order {\Aut {D_n} } = 2 \map \phi n$

where:

- $\order {\, \cdot \,}$ denotes the order of a group
- $\map \phi n$ is the Euler $\phi$ function.

## Proof

## Sources

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