Dihedral Group

The dihedral group of order 2n, usually written Dn, is the group of symmetries of a regular polygon with n sides, consisting of n rotations, including the identity, together with n reflections, so that combining any two of these symmetries by first performing one and then the other always yields another symmetry in the same group. It is one of the simplest and most commonly used examples in the teaching of group theory, since its elements and the multiplication rules between them can be visualized directly by physically rotating and flipping a paper polygon, making it a standard first example of a non-abelian group, a group in which the order of combining two elements matters. Dihedral groups appear throughout mathematics, physics, and chemistry wherever an object with polygonal symmetry needs to be described, including in crystallography, where the symmetry groups of many crystal structures are dihedral, and in the study of frieze and wallpaper patterns. The infinite dihedral group, an analogous but unbounded structure describing the symmetries of an infinite strip of repeating pattern, extends the same idea beyond any single finite polygon.