In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group 
acts on a set 
(this process is called a group
action), it permutes the elements of 
. Any particular element 
moves around in a fixed path which is called its orbit. In
the notation of set theory, the group orbit of a group element 
can be defined as
 |
(1)
|
where 
runs over all elements of the group
. For example, for the permutation
group 
,
the orbits of 1 and 2 are 
and the orbits of 3 and 4 are 
.
A group fixed point is an orbit consisting of a single element, i.e., an element that is sent to itself under all elements of the
group. The stabilizer of an element 
consists of all the permutations of 
that produce group fixed points
in 
, i.e., that send 
to itself. The stabilizers of 1 and 2 under 
are therefore 
, and the stabilizers of 3 and 4 are 
.
Note that if 
then 
, because 
iff 
. Consequently, the orbits partition 
and, given a permutation
group 
on a set 
,
the orbit of an element 
is the subset of 
consisting of elements to which some element 
can send 
.
For example, consider the action by the circle group 
on the sphere 
by rotations along its axis. Then the north pole is an orbit,
as is the south pole. The equator is a one-dimensional
orbit, as is a general orbit, corresponding to a line of latitude.
Orbits of a Lie group action may look different from each other. For example, 
,
the orthogonal group of signature 
, acts on the plane. It has three
different kinds of orbits: the origin (a group fixed
point, the four rays 
,
and the hyperbolas such as 
.
In general, an orbit may be of any dimension, up to the dimension of the Lie
group. If the Lie group 
is compact, then its orbits
are submanifolds.
The group's action on the 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,
 |
(2)
|
where 
is the orbit of 
in 
and 
is the stabilizer of 
in 
. This immediately gives the identity
 |
(3)
|
where 
denotes the order of group 
(Holton and Sheehan 1993, p. 27).
