A versor is a multivector that can be expressed as the geometric product of a number of non-null 1-vectors. That is, a versor \(\boldsymbol {V}\) can be written as \(\boldsymbol {V}=\prod _{i=1}^k \boldsymbol {n}_i\), where \(\left \{\boldsymbol {n}_1, \ldots , \boldsymbol {n}_k\right \} \subset \mathbb {G}_{p, q}^{\varnothing 1}\) with \(k \in \mathbb {N}^{+}\), is a set of not necessarily linearly independent vectors.
The subset of versors of \(\mathbb {G}_{p, q}\) together with the geometric product, forms a group, the Clifford group, denoted by \(\mathfrak {G}_{p, q}\).
A versor \(\boldsymbol {V} \in \mathfrak {G}_{p, q}\) is called unitary if \(\boldsymbol {V}^{-1}=\tilde {\boldsymbol {V}}\), i.e. \(\boldsymbol {V} \widetilde {\boldsymbol {V}}=+1\).
The set of unitary versors of \(\mathfrak {G}_{p, q}\) forms a subgroup \(\mathfrak {P}_{p, q}\) of the Clifford group \(\mathfrak {G}_{p, q}\), called the pin group.
A versor \(\boldsymbol {V} \in \mathfrak {G}_{p, q}\) is called a spinor if it is unitary \((\boldsymbol {V} \tilde {\boldsymbol {V}}=1)\) and can be expressed as the geometric product of an even number of 1-vectors. This implies that a spinor is a linear combination of blades of even grade.
The set of spinors of \(\mathfrak {G}_{p, q}\) forms a subgroup of the pin group \(\mathfrak {P}_{p, q}\), called the spin group, which is denoted by \(\mathfrak {S}_{p, q}\).