Define the set of symmetries of a rhombus as \[\mathscr{S}=\{e, R_{\text{ }O,\text{ }180^\circ}, r_{m_1}, r_{m_1}\}.\]

Claim \(\mathscr{S}\) is a group under composition.

0. Closure:

We want to show that comosing two symmetries equals a symmetry.

Let \(\square 1234\) be a non-square rhombus, and suppose F and G are in \(\mathscr{S}\).

\[F\circ G(\square 1234)=F(G(\square 1234))=F(\square 1234)=\square 1234\]

For example:

Let \(F=r_{m_1}\) and \(G=r_{m_2}\). Then \(F\circ G(\square 1234)=R_{O,180^\circ}\).

img.

1. Associative:

\(F\circ(G\circ H)=(F\circ G)\circ H\)

\[\begin{equation}\label{D14,1} \begin{split} F\circ(G\circ H) &= F\circ (G\circ H)(\square 1234)\\ &= F(G\circ H (\square 1234))\\ &= F(G(H(\square 1234)))\\ &= (F\circ G)\circ H(\square 1234) \end{split} \end{equation}\]

2. Identity:

e

Example: \(R_{O,180^\circ}\circ e=R_{O,180^\circ}\)

3. Inverses:

Every symmetry of a rhombus undoes itself.

\(e\circ e=e\)

\(R_{O,180^\circ}\circ R_{O,180^\circ}=e\)

\(r_{m_1}\circ r_{m_1}=e\)

\(r_{m_2}\circ r_{m_2}=e\)