of a group on a set is free if for every and every , the equation implies . Equivalently, an action is free when for any pair of elements , there is at most one group element such that .
Any group acts freely on itself by multiplication , which is called the (left) regular representation of .
For any set equipped with a transitive action , the group of -equivariant automorphisms of (i.e., bijections commuting with the action of ) acts freely on . In particular, suppose is such that for some , and let be arbitrary. By the assumption that acts transitively, there is a such that . But then -equivariance implies that . Since this holds for all , must be equal to the identity , and therefore acts freely on .