Symmetry Group of Regular Hexagon/Examples/Subgroup that Permutes A, C, E

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

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


 * SymmetryGroupRegularHexagon.png

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.

The set of elements of $D_6$ which permute vertices $A$, $C$ and $E$ form a subgroup of $D_6$ which is isomorphic to the dihedral group $D_3$.

Proof
Recall the group action of $D_6$ upon the vertices of $P$:

Let $H$ be the subset of elements of $D_6$ which permute vertices $A$, $C$ and $E$.

It is seen by inspection that $H$ is:
 * $H = \set {e, \alpha^2, \alpha^4, \beta, \alpha^2 \beta, \alpha^4 \beta}$

Setting $\gamma = \alpha^2$ we see that $H$ can be written:
 * $H = \set {e, \gamma, \gamma^2, \beta, \gamma \beta, \gamma^2 \beta}$

which is seen to be $D^3$.