User:D.Lazard/Double group

In physical sciences, the symmetry group of a physical body contains generally a subgroup (generally finite) of the group $SO(3)$ of the 3D rotation group. It may occur that the group $\{±1\}$ with two elements acts also on the body; this is typically the case in magnetohydrodynamics 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. This group extension is called a double group. This implies that two diffrent elements of the double group induce the spatial identity, and that a rotation of $2\pi$ may correspond to an element of the double group that is not the identity.

The classification of the finite double groups and their character tables is therefore physically meaningful and is thus the main part of the theory of double groups.