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A concept of symmetry is defined for general nonlinear control systems. It is shown, under various technical conditions, that nonlinear control systems with symmetries admit local and/or global decompositions in terms of lower dimensional subsystems and feedback loops. The structure of the individual subsystems is dependent on the structure of the symmetry group; for example, if the symmetry group is Abelian, one of the subsystems is a quadrature. An additional feature of the decomposition is that the state-space dimensions of the subsystems sum to the state-space dimension of the original system.