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Group Orbit


In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group G acts on a set X (this process is called a group action), it permutes the elements of X. Any particular element X moves around in a fixed path which is called its orbit. In the notation of set theory, the group orbit of a group element x can be defined as

 G(x)={gx in X:g in G},
(1)

where g runs over all elements of the group G. For example, for the permutation group G_1={(1234),(2134),(1243),(2143)}, the orbits of 1 and 2 are {1,2} and the orbits of 3 and 4 are {3,4}.

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 x consists of all the permutations of G that produce group fixed points in x, i.e., that send x to itself. The stabilizers of 1 and 2 under G_1 are therefore {(1234),(1243)}, and the stabilizers of 3 and 4 are {(1234),(2134)}.

Note that if y in G(x) then x in G(y), because y=gx iff x=g^(-1)y. Consequently, the orbits partition X and, given a permutation group G on a set S, the orbit of an element s in S is the subset of S consisting of elements to which some element G can send s.

OrbitGroup

For example, consider the action by the circle group S^1 on the sphere S^2 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, O(1,1), the orthogonal group of signature (1,1), acts on the plane. It has three different kinds of orbits: the origin (a group fixed point, the four rays {(+/-t,+/-t),t>0}, and the hyperbolas such as y^2-x^2=1. In general, an orbit may be of any dimension, up to the dimension of the Lie group. If the Lie group G is compact, then its orbits are submanifolds.

The group's action on the orbit through x 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,

 G(x)∼G/G_x,
(2)

where G(x) is the orbit of x in G and G_x is the stabilizer of x in G. This immediately gives the identity

 |G|=|G_x||G(x)|,
(3)

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


See also

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

Referenced on Wolfram|Alpha

Group Orbit

Cite this as:

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

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