Principal Homogeneous Space

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 action of G on X. An analogous definition holds in other categories. In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. Topological groups allow one to study the notion of continuous symmetries in the form of continuous group actions.