A group action ϕ:G×X->X is called faithful if there are no group elements g (except the identity element) such that g x = x for all x element X. Equivalently, the map ϕ induces an injection of G into the symmetric group S_X. So G can be identified with a permutation subgroup. Most actions that arise naturally are faithful. An example of an action which is not faithful is the action e^(i(x + y)) of G = R^2 = {(x, y)} on X = S^1 = {e^(i θ)}, i.e., ϕ(x, y, e^(i θ)) = e^(i(θ + x + y)).

Ado's theorem | effective action | free action | group | group orbit | Iwasawa's theorem | Lie group quotient space | transitive