Hasse Diagram/Examples/Subgroups of Symmetry Group of Rectangle

Example of Hasse Diagram

Consider the symmetry group of the rectangle:

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 $D_2$.

This Hasse diagram illustrates the subgroup relation on $\map D 2$.

Source of Name

This entry was named for Helmut Hasse.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Figure $12 \ (6)$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.1$