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,

• 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.