A faithful group action, also called an effective action, is a group action 
for which there are no group elements 
except the identity element such that 
for all 
. Equivalently, the map 
induces an injection
of 
into the symmetric group 
, so 
can be identified with a permutation
group.
The kernel of the action is 
, where 
is the isotropy group
at 
. Faithfulness is therefore equivalent to 
, where 
is the identity element
of 
.
An example of a group action which is not faithful is the action 
of 
on 
, i.e., 
.
It is possible for a Lie group 
to have a faithful group action on a smaller dimensional space 
. For a connected compact manifold 
with a smooth structure,
 |
is finite and is called the degree of symmetry of 
(Wiemeler 2013, p. 277).
