Symmetry Group of Regular Hexagon/Examples/Normalizer of Subgroup of Rotations

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.

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

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

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

Proof

We have that:

$\order {\gen \alpha} = 6 = \dfrac {\order {D_6} } 2$

and so:

$\index {D_6} {\gen \alpha} = 2$

From Subgroup of Index 2 is Normal, $\gen \alpha$ is normal in $D_6$.

The result follows from Normal Subgroup iff Normalizer is Group.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 48 \alpha$