Symmetry Group of Regular Hexagon/Examples/Center

Examples of Operations on Symmetry Group of Regular Hexagon
Let $\mathcal H = ABCDEF$ be a regular hexagon.

Let $D_6$ denote the symmetry group of $\mathcal H$.


 * SymmetryGroupRegularHexagon.png

Let $e$ denote the identity mapping

Let $\alpha$ denote rotation of $\mathcal H$ anticlockwise through $\dfrac \pi 3$ radians ($60 \degrees$)

Let $\beta$ denote reflection of $\mathcal H$ in the $AD$ axis.

The center of $D_6$ is:
 * $\map Z {D_6} = \set {e, \alpha^3}$

Proof
From Center of Dihedral Group:
 * $\map Z {D_n} = \begin{cases} e & : n \text { odd} \\ \set {e, \alpha^{n / 2} } & : n \text { even} \end{cases}$

for $n \ge 3$.

Here we have that $n = 6$ and so even.

Hence the result.