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 $\HH$.

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 $P$:

$D_6$ acts on the vertices of $\HH$ 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}$

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 $D^3$.

$\blacksquare$

Sources

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