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)
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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)
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where is the orbit of in and is the stabilizer of in . This immediately gives the identity
(3)
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where denotes the order of group (Holton and Sheehan 1993, p. 27).