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\section{Dihedral groups}

The dihedral group $D_n$ is the symmetry group of a regular $n$-sided polygon.  Generated by a rotation $r$ of $2\pi/n$ radians and by a mirroring operation $s$, there are $2n$ elements in the group $D_n$:

\begin{equation}
D_n = \{e, r, r^2, ..., r^{n-1}, s, rs, ..., r^{n-1}s\}
\end{equation}

The largest (proper) subgroup of a dihedral group $D_n$ is of course the group generated by just $r$, the cyclic group $C_n$, of index 2 inside $D_n$.


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