Symmetry Group of Regular Hexagon/Examples/Normalizer of Reflection
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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 normalizer of $\alpha$ is:
- $\map {N_{D_6} } {\set \beta} = \set {e, \beta, \alpha^3, \alpha^3 \beta}$
Proof
From Normalizer of Reflection in Dihedral Group:
- $\map {N_{D_n} } {\set \alpha} = \set {e, \beta, \alpha^{n / 2}, \alpha^{n / 2} \beta}$
for even $n$.
Hence the result by setting $n = 6$.
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 48 \alpha$