Orbits and stabilizers are closely related. For a fixed x in X, consider the map f : G → X given by g ↦ g ⋅ x. By definition the image f (G) of this map is the orbit G ⋅ x. The condition for two elements to have the same image is f ( g ) = f ( h ) ⟺ ⟺ g ⋅ ⋅ x = h ⋅ ⋅ x ⟺ ⟺ g − − 1 h ⋅ ⋅ x = x ⟺ ⟺ g − − 1 h ∈ ∈ G x ⟺ ⟺ h ∈ ∈ g G x . In other words, f (g) = f (h) if and only if g and h lie in the same coset for the stabilizer subgroup G x. Thus, the fiber f ({ y }) of f over any y in G ⋅ x is contained in such a coset, and every such coset also occurs as a fiber. Therefore f induces a bijection between the set G / G x of cosets for the stabilizer subgroup and the orbit G ⋅ x, which sends gG x ↦ g ⋅ x. This result is known as the orbit-stabilizer theorem.