In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). That is, there exists a short exact sequence of Lie groups

<math>1\to\mathbb{Z}_2\to\operatorname{Spin}(n)\to\operatorname{SO}(n)\to 1<math>

For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n, R). As a Lie group Spin(n) therefore shares its dimension <math>n(n-1)/2<math> and its Lie algebra with the special orthogonal group.

Spin(n) can be constructed as a subgroup of the invertible elements in the Clifford algebra C(n).

See also: spinor, spinor bundle, anyon