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

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,

where is the group orbit of in and is the stabilizer of in . This immediately gives the identity

where denotes the group order of group (Holton and Sheehan 1993, p. 27).

See also

Arc-Transitive Graph, Edge-Transitive Graph, Faithful Group Action, Free Action, Group, Group Fixed Point, Group Representation, Isotropy Group, Lie Group Quotient Space, Matrix Group, Permutation Group, Stabilizer, Topological Group, Transitive Group, Transitive Group Action, Vertex-Transitive Graph

Portions of this entry contributed by Todd Rowland

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References

Fraleigh, J. B. A First Course in Abstract Algebra, 7th ed. Reading, MA: Addison-Wesley, 2002.Holton, D. A. and Sheehan, J. The Petersen Graph. Cambridge, England: Cambridge University Press, pp. 26-27, 1993.Scott, W. R. "Cyclic Groups." §2.4 in Group Theory. New York: Dover, p. 255, 1987.

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Group Orbit

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Group Orbit." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupOrbit.html

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