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 $H := \set {e, r}$.

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 $\struct {\Z_2, +_2} \times \struct {\Z_2, +_2}$.