Dihedralgruppe

The Dihedral group, denoted as Dn or D2n, represents the symmetries of a regular n-sided polygon. These symmetries include rotations and reflections. For a regular n-gon, there are n rotational symmetries, including the identity (rotation by 0 degrees). Additionally, there are n reflectional symmetries. The total number of elements in the dihedral group Dn is 2n.

The group operation is composition, meaning applying one symmetry after another. The dihedral group is non-abelian for n > 2, meaning the order in which symmetries are applied matters. For instance, reflecting a square across its horizontal midline and then rotating it by 90 degrees clockwise results in a different final orientation than rotating it by 90 degrees clockwise first and then reflecting it horizontally.

The dihedral group Dn can be generated by two elements: a rotation 'r' by 2π/n radians and a reflection 's'. The relations that define the group are r^n = e (where 'e' is the identity), s^2 = e, and srs = r^-1. These relations capture all the possible combinations of rotations and reflections. Dihedral groups are fundamental in various areas of mathematics, including abstract algebra, geometry, and crystallography. They provide a concrete example of a finite group and are often used to illustrate group-theoretic concepts.