# Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a **homogeneous space** for a group *G* is a non-empty manifold or topological space *X* on which *G* acts transitively. The elements of *G* are called the **symmetries** of *X*. A special case of this is when the group *G* in question is the automorphism group of the space *X* – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, *X* is homogeneous if intuitively *X* looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of *G* be faithful (non-identity elements act non-trivially), although the present article does not. Thus there is a group action of *G* on *X* which can be thought of as preserving some "geometric structure" on *X*, and making *X* into a single *G*-orbit.