# Order of Dihedral Group

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## Theorem

The dihedral group $D_n$ is of order $2 n$.

## Proof

By definition, $D_n$ is the symmetry group of the regular polygon of $n$ sides.

Let $P$ be a regular $n$-gon.

By inspection, it is seen that:

- $(1): \quad$ there are $n$ symmetries of the vertices of $P$ by rotation
- $(2): \quad$ there are a further $n$ symmetries of the vertices of $P$ by rotation after reflected in any of the axes of symmetry of $P$.

Hence the result.

$\blacksquare$

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \theta$