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the Clifford algebra. This formula also defines an action of the Clifford group on the vector space ''V'' that preserves the norm ''Q'', and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements ''r'' of ''V'' of nonzero norm, and these act on ''V''
by the corresponding reflections that take ''v'' to ''v''−<''v'',''r''>''r'' / ''Q''(''r'') (In characteristic 2 these are called orthogonal transvections rather than reflections.)
Many authors define a slightly different Clifford group, by replacing
the action xvα(x)<sup>−1</sup> by xvx<sup>−1</sup>.
This alternative definition has several minor disadvantages in odd dimensions: for example, the map from the Clifford group to the orthogonal group is no longer onto, and the kernel is a little more complicated to describe.
The Clifford group Γ is the disjoint union of two subsets Γ<sub>0</sub> and Γ<sub>i</sub>, where Γ<sub>''i''</sub>
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also called the spinor norm. The spinor norm of the reflection of a vector
''r'' has image ''Q''(''r'') in ''F''<sup>*</sup>/''F''<sup>*2</sup>, and this property uniquely defines it on the orthogonal group.
== Spin and Pin groups ==
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