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

*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 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.