Let "X" be a non-empty set and "G" a group. Then "X" is called a "G"-space if it is equipped with an action of "G" on "X". Note that automatically "G" acts by automorphisms (bijections) on the set. If "X" in addition belongs to some category, then the elements of "G" are assumed to act as automorphisms in the same category. That is, the maps on "X" coming from elements of "G" preserve the structure associated with the category (for example, if X is an object in Diff then the action is required to be by diffeomorphisms). A homogeneous space is a "G"-space on which "G" acts transitively.