A group action is called free if, for all , implies (i.e., only the identity
element fixes *any* ). In other words, is free if the map sending to is injective, so that
implies for all . This means that all stabilizers
are trivial. A group with free action is said to act freely.

The basic example of a free group action is the action of a group on itself by left multiplication . As long as the group has more than the identity element, there is no element which satisfies for all . An example of a free action which is not transitive is the action of on by , which defines the Hopf map.