Formal Definition
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. Thus the maps on X effected by G are structure preserving. A homogeneous space is a G-space on which G acts transitively.
Succinctly, if X is an object of the category C, then the structure of a G-space is a homomorphism:
into the group of automorphisms of the object X in the category C. The pair (X,ρ) defines a homogeneous space provided ρ(G) is a transitive group of symmetries of the underlying set of X.
Read more about this topic: Homogeneous Space
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