Construction
The two pin groups correspond to the two central extensions
The group structure on Spin(V) (the connected component of determinant 1) is already determined; the group structure on the other component is determined up to the center, and thus has a ±1 ambiguity.
The two extensions are distinguished by whether the preimage of a reflection squares to ±1 ∈ Ker (Spin(V) → SO(V)), and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in O(V), r2 = 1, so the square of the preimage of a reflection (which has determinant one) must be in the kernel of Spin±(V) → SO(V), so, and either choice determines a pin group (since all reflections are conjugate by an element of SO(V), which is connected, all reflections must square to the same value).
Concretely, in Pin+, has order 2, and the preimage of a subgroup {1, r} is C2 × C2: if one repeats the same reflection twice, one gets the identity.
In Pin−, has order 4, and the preimage of a subgroup {1, r} is C4: if one repeats the same reflection twice, one gets "a rotation by 2π"—the non-trivial element of Spin(V) → SO(V) can be interpreted as "rotation by 2π" (every axis yields the same element).
Read more about this topic: Pin Group
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