A group orbit is the set of elements of a set 
obtainable from a given element 
by a group
action of a group 
. In the notation of set
theory, it is defined by
 |
(1)
|
where 
runs over all elements
of the group 
. In celestial mechanics, the path
a planet traces as it moves around the sun is called an orbit. For example, for the
permutation group, written in one-line notation,
, the
group orbits of 1 and 2 are 
and the group orbits of 3 and 4 are 
.
A group fixed point is a group 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 elements
of 
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 group orbits partition 
and, given a permutation
group 
on a set 
, the group orbit of an element 
is the subset
of 
consisting of elements
to which some element of 
can send 
.
For example, consider the group action by the circle group 
on the sphere 
by rotations about its axis.
Then the north pole is a group orbit, as is the south pole. The equator is a one-dimensional
group orbit, as is a general group orbit, corresponding to a circle
of constant latitude.
Group orbits of a smooth group action of a Lie group on a smooth manifold may look different
from each other. For example, 
, the orthogonal group
of matrix signature 
, acts on the plane. It has three
different kinds of group orbits: the origin (a group
fixed point), the union of the four rays 
(the light
cone with its vertex removed), and the hyperbolas 
for 
. In general, a group orbit may be of any dimension,
up to the dimension of the Lie
group. If the Lie group 
is compact, then its group
orbits are submanifolds.
The group action on the 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 group
orbit,
 |
(2)
|
where 
is the group orbit of 
in 
and 
is the stabilizer of 
in 
. This immediately gives the identity
 |
(3)
|
where 
denotes the group order of group 
(Holton and Sheehan 1993, p. 27).
