, an orbit
is a concept in group theory
. Consider a group G acting
on a set X
. The orbit
of an element 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
The orbits of a group action are the equivalence classes of the equivalence relation
defined by x
~ y iff
there exists g
. 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.