Dihedraalarühmadesse

Dihedraalarühmadesse, also known as the dihedral group $D_n$ or $D_{2n}$, represents the symmetries of a regular n-sided polygon. These symmetries consist of rotations and reflections. For a regular n-gon, there are n possible rotations, including the identity rotation (rotation by 0 degrees). The rotations are by $2\pi k / n$ radians, where k is an integer from 0 to $n-1$.

In addition to rotations, there are n reflections. Each reflection can be visualized as a flip across a line passing through the center of the polygon. These lines of reflection can either pass through opposite vertices (if n is even) or through a vertex and the midpoint of the opposite side (if n is odd). The combination of these n rotations and n reflections forms the dihedral group $D_n$, which has a total of 2n elements.

The group operation in a dihedral group is composition of these symmetries. For example, performing a rotation followed by a reflection results in another symmetry of the polygon, which can be expressed as a single rotation or a single reflection within the group. The dihedral group is non-abelian for $n > 2$, meaning the order in which symmetries are applied matters. $D_n$ is a fundamental example in group theory and appears in various areas of mathematics and science, including crystallography and the study of molecules.