Finite subgroups of SU(2)

Summary

In applied mathematics, finite subgroups of the special unitary group $\operatorname{SU}(2)$ are groups composed of rotations and related transformations, employed particularly in the field of physical chemistry. The symmetry group of a physical body generally contains a subgroup (typically finite) of the 3D rotation group. It may occur that the group $\{\pm1\}$ with two elements acts also on the body; this is typically the case in magnetism for the exchange of north and south poles, or in quantum mechanics for the change of spin sign. In this case, the symmetry group of a body may be a central extension of the group of spatial symmetries by the group with two elements. Hans Bethe introduced the term "double group" (Doppelgruppe) for such a group, in which two different elements induce the spatial identity, and a rotation of $2\pi$ may correspond to an element of the double group that is not the identity.