Symmetry Group of Regular Hexagon/Examples/Subgroup that Fixes C

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 $\mathcal H$.

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 operations of $D_6$ that fix vertex $C$ form a subgroup of $D_6$ which is isomorphic to the parity group.

Proof

$D_6$ acts on the vertices of $\mathcal H$ according to this table:

$\begin{array}{cccccccccccc} e & \alpha & \alpha^2 & \alpha^3 & \alpha^4 & \alpha^5 & \beta & \alpha \beta & \alpha^2 \beta & \alpha^3 \beta & \alpha^4 \beta & \alpha^5 \beta \\ \hline A & B & C & D & E & F & A & B & C & D & E & F \\ B & C & D & E & F & A & F & A & B & C & D & E \\ C & D & E & F & A & B & E & F & A & B & C & D \\ D & E & F & A & B & C & D & E & F & A & B & C \\ E & F & A & B & C & D & C & D & E & F & A & B \\ F & A & B & C & D & E & B & C & D & E & F & A \\ \end{array}$

It is seen by inspection that the only elements of $D_6$ which fix $C$ are $e$ and $\alpha^4 \beta$.

It is further seen that $\alpha^4 \beta$ is the reflection whose axis is $CF$.

The Cayley table of these $2$ elements can be shown to be:

$\begin{array}{c|cc} & e & \alpha^4 \beta \\ \hline e & e & \alpha^4 \beta \\ \alpha^4 \beta & \alpha^4 \beta & e \\ \end{array}$

whose isomorphism to the parity group is immediate.

$\blacksquare$