In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial. Equivalently, a principal homogeneous space for a group G is a set X on which G acts freely and transitively, so 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. （以下略）
