A group action G×X->X is called free if, for all x element X, g x = x implies g = I (i.e., only the identity element fixes any x). In other words, G×X->X is free if the map G×X->X×X sending (g, x) to (a(g, x), x) is injective, so that a(g, x) = x implies g = I for all g, x. This means that all stabilizers are trivial. A group with free action is said to act freely.

effective action | group | group orbit | group representation | isotropy group | Lie group quotient space | matrix group | stabilizer | topological group | transitive group action