A free action is a group action 
such that, for
all 
,
implies 
(i.e., only the identity
element fixes any 
). Equivalently, 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 
. For each 
, the equation 
implies 
after right multiplication
by 
. An example of a free action which
is not transitive is the action
of 
on 
by 
, which
defines the Hopf map.
On a nonempty set, every free action is a faithful group action, but the converse fails. The natural group
action of the symmetric group 
on three points is faithful,
whereas the transposition 
fixes the third point, so this
group action is not free.
For a smooth group action of a Lie group on a smooth manifold, freeness together
with a proper group action gives the quotient
space a smooth manifold structure, and the
quotient projection is a principal bundle (Koster
n.d., p. 2). Freeness alone does not imply properness. For example, the group 
with the discrete
topology acts freely on the unit circle by 
for an irrational
number 
.
The map 
from 
to 
cannot be a proper map,
since the inverse image of its entire compact
target is the noncompact space 
.
