Symmetry Group of Rectangle/Cayley Table

Cayley Table of Symmetry Group of Rectangle

Definition

Let $\RR = ABCD$ be a (non-square) rectangle.

The various symmetries 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:

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.11$