Symmetry Group of Regular Hexagon/Examples/Normalizer of Reflection

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 normalizer of $\alpha$ is:
 * $\map {N_{D_6} } {\set \beta} = \set {e, \beta, \alpha^3, \alpha^3 \beta}$

Proof
From Normalizer of Reflection in Dihedral Group:
 * $\map {N_{D_n} } {\set \alpha} = \set {e, \beta, \alpha^{n / 2}, \alpha^{n / 2} \beta}$

for even $n$.

Hence the result by setting $n = 6$.