Symmetry Group of Rhombus/Cayley Table

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Cayley Table of Symmetry Group of Rhombus

Definition

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

SymmetryGroupRhombus.png

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.


Cayley Table

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

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