Definition:Symmetry Group of Regular Hexagon

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Group Example

Let $\mathcal H = ABCDEF$ be a regular hexagon.

SymmetryGroupRegularHexagon.png

The various symmetry mappings of $\mathcal H$ are:

The identity mapping $e$
The rotations through multiples of $60 \degrees$
The reflections in the indicated axes.

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 symmetries of $\mathcal H$ form the dihedral group $D_6$.


Group Action on Vertices

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

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


Examples

Subgroup of Operations that Fix $C$

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


Subgroup of Operations that Permute $A$, $C$ and $E$

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 $D_3$.


Subgroup of Operations Generated by $\alpha^4$ and $\alpha^3 \beta$

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:

$H \cong D_3$


Elements of Form $\beta \alpha^k$ in Form $\alpha^i \beta^j$

Consider elements of $D_6$ of the form $\beta \alpha^k$, for $k \in \Z^{\ge 0}$.

They can be expressed in the form:

$\beta \alpha^k = \alpha^{6 - k} \beta$


Center

The center of $D_6$ is:

$\map Z {D_6} = \set {e, \alpha^3}$


Normalizer of $\alpha$

The normalizer of $\alpha$ is:

$\map {N_{D_6} } {\set \alpha} = \set {e, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5}$


Normalizer of $\beta$

The normalizer of $\alpha$ is:

$\map {N_{D_6} } {\set \beta} = \set {e, \beta, \alpha^3, \alpha^3 \beta}$


Normalizer of $\gen \alpha$

Let $\gen \alpha$ denote the subgroup generated by $\alpha$.

The normalizer of $\gen \alpha$ is $D_6$ itself:

$\map {N_{D_6} } {\gen \alpha} = D_6$


Sources