Rotasjonsundergruppen

Rotasjonsundergruppen, also known as the rotation subgroup, is a concept in group theory, a branch of mathematics that studies the properties of symmetry. It refers to the set of all rotations that map a given object onto itself. In other words, it is the subgroup of the group of symmetries of an object that consists only of rotations.

The rotation subgroup is particularly important in the study of symmetry in two and three-dimensional space. For example, in two dimensions, the rotation subgroup of a regular polygon is the set of rotations that map the polygon onto itself. Similarly, in three dimensions, the rotation subgroup of a regular polyhedron is the set of rotations that map the polyhedron onto itself.

The order of the rotation subgroup is equal to the number of distinct rotations that map the object onto itself. For example, the rotation subgroup of a regular tetrahedron has order 3, as there are three distinct rotations that map the tetrahedron onto itself.

The rotation subgroup is a normal subgroup of the full group of symmetries of an object. This means that it is closed under conjugation, which is a property that makes it easier to study and understand.

In summary, the rotation subgroup is a fundamental concept in group theory that plays a crucial role in the study of symmetry. It is the set of all rotations that map a given object onto itself, and it is a normal subgroup of the full group of symmetries of the object.