Internal Direct Product Theorem/Examples/Symmetry Group of Rectangle
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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 $13.2$