Symmetry Group of Regular Hexagon/Examples/Elements of Form beta alpha^k

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.

Consider elements of $D_6$ of the form $\beta \alpha^k$, for $k \in \Z^{\ge 0}$.

They can be expressed in the form:
 * $\beta \alpha^k = \alpha^{6 - k} \beta$

Proof
From Product of Generating Elements of Dihedral Group:
 * $\beta \alpha^k = \alpha^{n - k} \beta$

for the dihedral group $D_n$.

The result follows by setting $n = 6$.