Every Clifford algebra \(\mathrm {Cl}(\Phi )\) possesses a canonical anti-automorphism \(t: \mathrm {Cl}(\Phi ) \rightarrow \mathrm {Cl}(\Phi )\) satisfying the properties
\[
t(x y)=t(y) t(x), \quad t \circ t=\mathrm {id}, \quad \text { and } \quad t(i(v))=i(v),
\]
for all \(x, y \in \mathrm {Cl}(\Phi )\) and all \(v \in V\). Furthermore, such an anti-automorphism is unique.
Every Clifford algebra \(\mathrm {Cl}(\Phi )\) has a unique canonical automorphism \(\alpha : \mathrm {Cl}(\Phi ) \rightarrow \mathrm {Cl}(\Phi )\) satisfying the properties
\[
\alpha \circ \alpha =\mathrm {id}, \quad \text { and } \quad \alpha (i(v))=-i(v),
\]
for all \(v \in V\).
First, we define conjugation on a Clifford algebra \(\mathrm {Cl}(\Phi )\) as the map
\[
x \mapsto \bar {x}=t(\alpha (x)) \text { for all } x \in \mathrm {Cl}(\Phi ) .
\]
Given a finite dimensional vector space \(V\) and a quadratic form \(\Phi \) on \(V\), the Clifford group of \(\Phi \) is the group
\[
\Gamma (\Phi )=\left \{x \in \mathrm {Cl}(\Phi )^* \mid \alpha (x) v x^{-1} \in V \quad \text { for all } v \in V\right \} .
\]
The map \(N: \mathrm {Cl}(Q) \rightarrow \mathrm {Cl}(Q)\) given by
\[
N(x)=x \bar {x}
\]
is called the norm of \(\mathrm {Cl}(\Phi )\).
We also define the group \(\Gamma ^{+}(\Phi )\), called the special Clifford group, by
\[
\Gamma ^{+}(\Phi )=\Gamma (\Phi ) \cap \mathrm {Cl}^0(\Phi ) .
\]
We define the pinor group \(\operatorname {Pin}(p, q)\) as the group
\[
\operatorname {Pin}(p, q)=\left \{x \in \Gamma _{p, q} \mid N(x)= \pm 1\right \},
\]
and the spinor group \(\operatorname {Spin}(p, q)\) as \(\operatorname {Pin}(p, q) \cap \Gamma _{p, q}^{+}\).
The restriction of \(\rho : \Gamma _{p, q} \rightarrow \mathbf {G L}(n)\) to the pinor group \(\operatorname {Pin}(p, q)\) is a surjective homomorphism \(\rho : \mathbf {P i n}(p, q) \rightarrow \mathbf {O}(p, q)\) whose kernel is \(\{-1,1\}\), and the restriction of \(\rho \) to the spinor group \(\mathbf {S p i n}(p, q)\) is a surjective homomorphism \(\rho : \mathbf {S p i n}(p, q) \rightarrow \mathbf {S O}(p, q)\) whose kernel is \(\{-1,1\}\).
Remark: According to Atiyah, Bott and Shapiro, the use of the name \(\operatorname {Pin}(k)\) is a joke due to Jean-Pierre Serre (Atiyah, Bott and Shapiro Clifford modules, page 1).