Symmetry Group of Regular Hexagon/Examples
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.
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:
- $\map {N_{D_6} } {\gen \alpha} = D_6$