Pyörähdysryhmät

Pyörähdysryhmät are mathematical groups that describe the rotational symmetries of geometric objects. In group theory, a pyörähdysryhmä is a subgroup of the group of all isometries of a space that consists solely of rotations. These groups are fundamental in the study of geometry, crystallography, and molecular symmetry.

A common example is the cyclic group of order n, denoted $C_n$. This group represents the rotations that leave a regular n-gon invariant. The elements of $C_n$ are rotations by angles of $0, 2\pi/n, 4\pi/n, \dots, 2\pi(n-1)/n$ radians around the center of the polygon. Applying these rotations in succession results in further rotations within the set. The composition of any two rotations in the set yields another rotation within the set, and the set includes the identity element (rotation by 0).

More complex pyörähdysryhmät can arise from objects with higher dimensions or more intricate symmetry. For instance, the rotational symmetry group of a cube is a finite group that includes rotations around axes passing through the centers of opposite faces, edges, and vertices. Understanding pyörähdysryhmät allows mathematicians and scientists to classify and analyze the symmetrical properties of various structures, from simple shapes to complex molecular arrangements. The abstract algebraic structure of these groups captures the essence of rotational invariance.