Free action of a group ($gx=x⟹g=1$)

A group action is free when only the identity fixes any given point, or equivalently every stabilizer subgroup is trivial. The trivial group acts freely on every space. A nontrivial group acting freely must act by a fixed-point-free map for each nonidentity element; the fixed-point property of even-dimensional complex projective space obstructs such actions. For a finite group acting freely on a Hausdorff manifold, the quotient map is a covering map.