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I am studying properties of manifolds with group action and in particular the structure of manifolds obtained from group action. We have an approach of discrete group action different from connected Lie group action. Free action on a manifold has strict restriction. However, a manifold has many group actions if its action is not free. I am interested in Smith's problem: Is two tangential representations isomorphic in a finite group action on a sphere with just two fixed points? There are many finite groups whose actions induce that two tangential representations are not isomorphic. Given two representations, when is there an action on a sphere with two fixed points such that two representations are isomorphic to the tangential representations over the fixed points?