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

Examples of Operations on Symmetry Group of Regular Hexagon

Let $\HH = ABCDEF$ be a regular hexagon.

Let $D_6$ denote the symmetry group of $\HH$.

Let $e$ denote the identity mapping

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

Let $\beta$ denote reflection of $\HH$ 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 $n = 6$.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35 \zeta \ (4)$