Internal Direct Product Theorem/Examples/Symmetry Group of Rectangle

Example of Use of Internal Direct Product Theorem

Consider the symmetry group of the rectangle $D_2$:

Let $\RR = ABCD$ be a (non-square) rectangle.

The various symmetry mappings of $\RR$ are:

The identity mapping $e$

The rotation $r$ (in either direction) of $180^\circ$

The reflections $h$ and $v$ in the indicated axes.

The symmetries of $\RR$ form the dihedral group $D_2$.

Let $H := \set {e, r}$.

Let $K := \set {e, h}$.

Then $H$ and $K$ are subgroups of $D_2$ which fulfil the conditions of the Internal Direct Product Theorem, as:

$r \circ h = v = h \circ r$

Thus $D_2$ is the internal group direct product of $H$ and $K$.

Both $H$ and $K$ are isomorphic to $\struct {\Z_2, +_2}$, the additive group of integers modulo $2$.

Hence by Isomorphism of External Direct Products:

$D_2$ is isomorphic to $\struct {\Z_2, +_2} \times \struct {\Z_2, +_2}$.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Example $13.2$