Definition:Symmetry Group of Rhombus
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Group Example
Let $\RR = ABCD$ be a (non-square) rhombus.
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) rhombus can be written:
- $\begin{array}{c|cccc}
& e & r & h & v \\
\hline e & e & r & h & v \\ r & r & e & v & h \\ h & h & v & e & r \\ v & v & h & r & e \\ \end{array}$
$D_2$ acts on the vertices of $\RR$ according to this table:
- $\begin{array}{cccc}
e & r & h & v \\
\hline A & C & A & C \\ B & D & D & B \\ C & A & C & A \\ D & B & B & D \\ \end{array}$
Also see
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Exercise $4 \ \text{(c)}$