Symmetry Group of Regular Hexagon/Examples/Normalizer of Rotation

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:

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

Proof

From Normalizer of Rotation in Dihedral Group:

$\map {N_{D_n} } {\set \alpha} = \gen \alpha$

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: $\S 48 \alpha$