Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively, meaning that for any x, y in X there exists a unique g in G such that x·g = y where · denotes the (right) action of G on X. An analogous definition holds in other categories where, for example,
If G is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions. To state the definition more explicitly, X is a G-torsor if X is nonempty and is equipped with a map (in the appropriate category) X × G → X such that
for all x ∈ X and all g,h ∈ G and such that the map X × G → X × X given by
is an isomorphism (of sets, or topological spaces or ..., as appropriate). Note that this means that X and G are isomorphic. However —and this is the essential point—, there is no preferred 'identity' point in X. That is, X looks exactly like G except that which point is the identity has been forgotten. This concept is often used in mathematics as a way of passing to a more intrinsic point of view, under the heading 'throw away the origin'.
Since X is not a group we cannot multiply elements; we can, however, take their "quotient". That is, there is a map X × X → G that sends (x,y) to the unique element g = x \ y ∈ G such that y = x·g.
The composition of this operation with the right group action, however, yields a ternary operation X × (X × X) → X × G → X that serves as an affine generalization of group multiplication and is sufficient to both characterize a principal homogeneous space algebraically, and intrinsically characterize the group it is associated with. If is the result of this operation, then the following identities
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