Rotation group SO(3)

The rotation group in the narrow sense is the special orthogonal group or all rotations in the real three-dimensional space (in case) or in the real plane (in case), in the latter case it is called a channel group. Its elements are the rotation matrices, ie orthogonal matrices with

And determinant one.

In addition, a subset of these groups is called a real rotation group of a two - or three-dimensional figure, if it includes all the rotations that map the figure onto itself, that is the subgroup of rotations in the symmetry group of the body or of the figure. To distinguish the full -dimensional rotation group is called.

In the further and figuratively be the orthogonal groups, which are subgroups of the real general linear group, whose members are unimodular orthogonal matrices, called rotation groups for higher dimensions as ( full ). More generally, is sometimes in any commutative ring with unity and a natural number, the group of rotations of the orthogonal group of a special module called.

Properties

Every real full rotation group is a Lie group. A topology she gets in a canonical way as a matrix group and thus also their Lie Rupp structure is defined. Each full rotation group is a compact topological group.

The Lie algebra of the. She is a real form of the Lie algebra. The latter results in the special unitary group defined on an overlay group of degree for.

Rotating groups of figures

The word rotation group is also used as a term for those sub-group of symmetries of a given geometric object that maps a planimetric figure or a stereometric body by turning on itself. Such a rotation group is then a (usually finite) subset of or and consists exactly of all those twists through which this figure and this body is transferred to yourself.

Examples

• In the plane

• In space