Order of Dihedral Group

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.


 * SymmetryGroupOddPolygon.png SymmetryGroupEvenPolygon.png

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.