---
tag: 0BZVNWJ352
kind: statement
role: proposition
name: Order of the dihedral group
msc: [20B25]
uses: [09GK3DE0WV, 0SEX9F9J5Z]
sources:
  - url: https://en.wikipedia.org/wiki/Dihedral_group
    locator: Definition
---

## Statement

Let $n \geq 3$.
The [[dihedral group|0JZ27DMJ5D]] $D_n$ has [[order|0B85S2V2DR]] $2n$.

## Remarks

A symmetry of the regular $n$-gon is determined by the image of one vertex and by whether it keeps or reverses the orientation.
There are $n$ choices for the image of the vertex and $2$ choices for the orientation.
For example, $D_3$ has order $6$ and $D_4$ has order $8$.

## Explanations

### L1 human

A symmetry can send a fixed vertex to any of the $n$ vertices.
After that, the polygon can be face up or face down.
So there are $n \cdot 2 = 2n$ symmetries.