Definition:Symmetry Group of Rectangle

Group Example

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:

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

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

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

Also see

• Definition:Klein Four-Group

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 44.5$ Some consequences of Lagrange's Theorem
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Exercise $4 \ \text{(a)}$