Symmetry Group of Rectangle/Cayley Table

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Cayley Table of Symmetry Group of 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 $D_2$.

Cayley Table

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

 & 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}$