Let 
be a permutation
group on a set 
and 
be an element of 
. Then
 |
(1)
|
is called the stabilizer of 
and consists of all the permutations of 
that produce group fixed points
in 
, i.e., that send 
to itself. 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 an 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).
