Abstract: This paper introduced the Mǒbius transformation group on Sⁿ, the Lorentz transformation group O⁺ (n+1, 1), conformal transformation group, the definition of protection goals and the nature of the transformation group; discussed the relationship among the four transformation groups; Pointed out that the four types of transformation group had the equivalent role on sⁿ, then the proof was given in four steps.