Definition:Symmetry Group of Rectangle

Group Example
Let $\mathcal R = ABCD$ be a (non-square) rectangle.


 * SymmetryGroupRectangle.png

The various symmetry mappings of $\mathcal R$ 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 $\mathcal R$ form the dihedral group $D_2$.

Its Cayley table can be written:


 * $\begin{array}{c|cccccc}

& 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}$

$D_2$ acts on the vertices of $\mathcal R$ according to this table:


 * $\begin{array}{cccccccccccc}

e & r & h & v \\ \hline A & C & D & B \\ B & D & C & A \\ C & A & B & D \\ D & B & A & C \\ \end{array}$