Symmetry Group of Regular Hexagon/Examples/Subgroup Generated by alpha^4 and alpha^3 beta

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.

Let $H$ be the subgroup of $D_6$ generated by $\alpha^4$ and $\alpha^3 \beta$.

Then:

$H = \set {e, \alpha^2, \alpha^4, \alpha \beta, \alpha^3 \beta, \alpha^5 \beta}$

and:

$H \cong D_3$

Proof

Demonstration by Cayley table:

$\begin{array}{c|cccccc}

 & e & \alpha^2 & \alpha^4 & \alpha \beta & \alpha^3 \beta & \alpha^5 \beta \\

\hline e & e & \alpha^2 & \alpha^4 & \alpha \beta & \alpha^3 \beta & \alpha^5 \beta \\ \alpha^2 & \alpha^2 & \alpha^4 & e & \alpha^3 \beta & \alpha^5 \beta & \alpha \beta \\ \alpha^4 & \alpha^4 & e & \alpha^2 & \alpha^5 \beta & \alpha \beta & \alpha^3 \beta \\ \alpha \beta & \alpha \beta & \alpha^5 \beta & \alpha^3 \beta & e & \alpha^4 & \alpha^2 \\ \alpha^3 \beta & \alpha^3 \beta & \alpha \beta & \alpha^5 \beta & \alpha^2 & e & \alpha^4 \\ \alpha^5 \beta & \alpha^5 \beta & \alpha^3 \beta & \alpha \beta & \alpha^4 & \alpha^2 & e \\ \end{array}$

It can be seen by inspection that this is $D_3$.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35 \zeta \ (3)$