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 ${\displaystyle -I.}$

Let ${\displaystyle V}$ be a vector space with a non-degenerate quadratic form ${\displaystyle Q}$. The pin group ${\displaystyle \operatorname {Pin} (V,Q)}$ is the subset of the Clifford algebra ${\displaystyle \operatorname {Cl} (V,Q)}$ consisting of elements of the form ${\displaystyle v_{1}v_{2}\cdots v_{k}}$, where the ${\displaystyle v_{i}}$ are vectors such that ${\displaystyle Q(v_{i})=\pm 1}$. The spin group ${\displaystyle \operatorname {Spin} (V,Q)}$ is defined similarly, but with ${\displaystyle k}$ restricted to be even; it is a subgroup of the pin group.

In this article, ${\displaystyle V}$ is always a real vector space. When ${\displaystyle V}$ has basis vectors ${\displaystyle e_{1},...,e_{p+q}}$ satisfying ${\displaystyle e_{1}^{2},...,e_{p}^{2}=1}$ and ${\displaystyle e_{p+1}^{2},...,e_{p+q}^{2}=-1,}$ the pin group is denoted Pin(p, q).

Geometrically, for vectors ${\displaystyle v}$ with ${\displaystyle Q(v)\neq 0}$, ${\displaystyle -vwv^{-1}}$ is the reflection of a vector ${\displaystyle w}$ across the hyperplane orthogonal to ${\displaystyle v}$. More generally, an element ${\displaystyle v_{1}v_{2}\cdots v_{k}}$ of the pin group acts on vectors by transforming ${\displaystyle w}$ to ${\displaystyle (-1)^{k}v_{1}v_{2}\cdots v_{k}w(v_{1}v_{2}\cdots v_{k})^{-1}}$, which is the composition of k reflections. Since every orthogonal transformation can be expressed as a composition of reflections (the Cartan–Dieudonné theorem), it follows that this representation of the pin group is a homomorphism from the pin group onto the orthogonal group. This is often called the twisted adjoint representation. The elements ±1 of the pin group are the elements which map to the identity ${\displaystyle I\in O(p,q)}$, and every element of ${\displaystyle O(p,q)}$ corresponds to exactly two elements of ${\displaystyle \operatorname {Pin} (p,q)}$.

The pin group of a definite form maps onto the orthogonal group, and each component is simply connected (in dimension 3 and higher): it double covers the orthogonal group. The pin groups for a positive definite quadratic form Q and for its negative −Q are not isomorphic, but the orthogonal groups are.

In terms of the standard forms, O(n, 0) = O(0, n), but Pin(n, 0) and Pin(0, n) are in general not isomorphic. Using the "+" sign convention for Clifford algebras (where ${\displaystyle v^{2}=Q(v)\in \mathrm {Cl} (V,Q)}$), one writes