Dihedral group ($D_n$)

For $n\ge 3$, the dihedral group is the symmetry group of a regular $n$-gon, including rotations and reflections. Using the convention that the subscript gives the group order,Its $2n$ elements are $r^j$ and $r^{j}s$, with $0\le j<n$: the first are rotations, the second reflections. The relation $sr=r^{-1}s$ reduces every word to one of these forms. Geometrically the image of one vertex and a choice of direction around the polygon distinguish all $2n$ forms.

Another common convention writes $D_n$ for this same group, indexing by polygon vertices. For example the symmetry group of a square has eight elements and may be called either $D_8$ or $D_4$. The group order or group presentation resolves that notation difference. Its rotation subgroup is a cyclic group of index two, and a reflection conjugates each rotation to its inverse.