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 , where, for example, G is a topological group, X is a topological space and the action is continuous, G is a Lie group, X is a smooth manifold and the action is smooth, G is an algebraic group, X is an algebraic variety and the action is regular. 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 or G-principal homogeneous space if X is nonempty and is equipped with a map (in the appropriate category) X × G → X such that x·1 = x x·(gh) = (x·g)·h 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, i.e. in the category in question). Note that this means that X and G are isomorphic (in the category in question; not as groups: see the following). 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.