Symmetry Group of Rectangle is Klein Four-Group

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Theorem

The symmetry group of the rectangle is the Klein $4$-group.


Proof

Comparing the Cayley tables of the symmetry group of the rectangle with the Klein $4$-group the isomorphism can be seen:


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

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


$\begin{array}{c|cccc} & e & a & b & c \\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end{array}$


Thus the required isomorphism $\phi$ can be set up as:

\(\ds \map \phi e\) \(=\) \(\ds e\)
\(\ds \map \phi r\) \(=\) \(\ds a\)
\(\ds \map \phi h\) \(=\) \(\ds b\)
\(\ds \map \phi v\) \(=\) \(\ds c\)

$\blacksquare$


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