An action
of a group on a set is called free if for every , the equation implies (the neutral element).
Equivalently, an action is free when for any pair of elements , there is at most one group element such that .
Beware the similarity to and difference with effective action: a free action is effective, but an effective action need not be free.
A free action that is also transitive is called regular.
Any group acts freely on itself by multiplication , which is called the (left) regular representation of .
An action of on a set corresponds to an arbitrary involution , but the action is free just in case is a fixed point-free involution.
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 .
A combinatorial species is said to be flat if all of the actions are free (see Combinatorial species and tree-like structures). For example, the species of linear orders is flat.
Last revised on November 23, 2020 at 12:14:40. See the history of this page for a list of all contributions to it.