Pin group

In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group.

In general the map from the Pin group to the orthogonal group is not surjective or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both.

The non-trivial element of the kernel is denoted ${\displaystyle -1,}$ which should not be confused with the orthogonal transform of reflection through the origin, generally denoted ${\displaystyle -I.}$