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.