# Definition:Symmetry Group of Parallelogram

## Group Example

Let $\PP = ABCD$ be a (non-rectangular) parallelogram.

The various symmetry mappings of $\PP$ are:

The identity mapping $e$
The rotation $r$ (in either direction) of $180^\circ$.

The symmetries of $\PP$ form the dihedral group $D_1$.

### Cayley Table

The Cayley table of the symmetry group of the (non-rectangular) parallelogram can be written:

$\begin{array}{c|cccc} & e & r \\ \hline e & e & r \\ r & r & e \\ \end{array}$