# Symmetry Group of Rectangle/Cayley Table

## Cayley Table of Symmetry Group of Rectangle

### Definition

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 $D_2$.

### Cayley Table

The Cayley table of the symmetry group of the (non-square) rectangle can be written:

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