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:
\(\ds \map \phi e\) | \(=\) | \(\ds e\) | ||||||||||||
\(\ds \map \phi r\) | \(=\) | \(\ds a\) | ||||||||||||
\(\ds \map \phi h\) | \(=\) | \(\ds b\) | ||||||||||||
\(\ds \map \phi v\) | \(=\) | \(\ds c\) |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 44$. Some consequences of Lagrange's Theorem: Illustration $1$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Klein's four group