As Topological Group
Every connected topological group has a unique universal cover as a topological space, which has a unique group structure as a central extension by the fundamental group. For a disconnected topological group, there is a unique universal cover of the identity component of the group, and one can take the same cover as topological spaces on the other components (which are principal homogeneous spaces for the identity component) but the group structure on other components is not uniquely determined in general.
The Pin and Spin groups are particular topological groups associated to the orthogonal and special orthogonal groups, coming from Clifford algebras: there are other similar groups, corresponding to other double covers or to other group structures on the other components, but they are not referred to as Pin or Spin groups, nor studied much.
Recently, Andrzej Trautman found the set of all 32 inequivalent double covers of O(p) x O(q), the maximal compact subgroup of O(p, q) and an explicit construction of 8 double covers of the same group O(p, q).
Read more about this topic: Pin Group
Famous quotes containing the word group:
“Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
—Simone Weil (1910–1943)