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Orbit (mathematics)

In mathematics, an orbit is a concept in group theory. Consider a group G acting on a set X. The orbit of an element x of X is the set of elements of X to which x can be moved by the elements of G; it is denoted by Gx. That is

<math> Gx = \left\{ g.x : g \in G \right\} </math>

The orbits of a group action are the equivalence classes of the equivalence relation on X defined by x ~ y iff there exists g in G with x = g.y. As a consequence, every element of X belongs to one and only one orbit.

If two elements x and y belong to the same orbit, then their stabilizer subgroups[?] Gx and Gy are isomorphic. More precisely: if y = g.x, then the inner automorphism of G given by h |-> ghg-1 maps Gx to Gy.

If both G and X are finite, then the size of any orbit is a factor of the order of the group G by the orbit-stabilizer theorem.

The set of all orbits is denoted by X/G. Burnside's lemma gives a formula that allows to calculate the number of orbits.

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