The stabilizer of a point 
under a group action of a
group 
on a set 
is the subgroup
 |
(1)
|
of elements that fix 
.
It is also called the isotropy group of 
. For example, the stabilizer of 1 and of 2 under the permutation
group 
is both 
,
and the stabilizer of 3 and of 4 is 
.
More generally, the subset of all images of 
under permutations of the group 
 |
(2)
|
is called the group orbit of 
in 
.
A group's action on a group orbit through 
is transitive, and
so is related to its isotropy group. In particular,
the cosets of the isotropy subgroup correspond to the elements in the orbit,
 |
(3)
|
where 
is the orbit of 
in 
and 
is the stabilizer of 
in 
.
This immediately gives the identity
 |
(4)
|
where 
denotes the order of group 
(Holton and Sheehan 1993, p. 27).
