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.
