= Free action of a group
{title2=$gx=x\Longrightarrow g=1$}
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>.
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