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[?] G_{x} and G_{y} are isomorphic. More precisely: if y = g.x, then the inner automorphism of G given by h > ghg^{1} maps G_{x} to G_{y}.
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 orbitstabilizer 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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