Let x and y be two elements in X, and let g be a group element such that y = g ⋅ x. Then the two stabilizer groups G x and G y are related by G y = gG x g. Proof: by definition, h ∈ G y if and only if h ⋅(g ⋅ x) = g ⋅ x. Applying g to both sides of this equality yields (g hg)⋅ x = x ; that is, g hg ∈ G x. An opposite inclusion follows similarly by taking h ∈ G x and x = g ⋅ y.