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
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.
See also: