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.

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: $\S 26 \theta$