Symmetry Group of Rectangle is Klein Four-Group

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:

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

The Klein $4$-group can be described completely by showing its Cayley table:

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

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