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''of a point  $  x $
relative to a [[Group|group]]  $  G $
acting on a set  $  X $(
on the left)''

The set

$$ 
G( x)  =  \{ {g( x) } : {g \in G } \}
.
$$

The set

$$ 
G _ {x}  =  \{ {g \in G } : {g( x) = x } \}
$$

is a subgroup in  $  G $
and is called the stabilizer or stationary subgroup of the point  $  x $
relative to  $  G $. 
The mapping  $  g \mapsto g( x) $, 
$  g \in G $, 
induces a bijection between  $  G/G _ {x} $
and the orbit  $  G( x) $. 
The orbits of any two points from  $  X $
either do not intersect or coincide; in other words, the orbits define a partition of the set  $  X $. 
The quotient by the equivalence relation defined by this partition is called the orbit space of  $  X $
by  $  G $
and is denoted by  $  X/G $. 
By assigning to each point its orbit, one defines a canonical mapping  $  \pi _ {X,G} :  X \rightarrow X/G $. 
The stabilizers of the points from one orbit are conjugate in  $  G $, 
or, more precisely,  $  G _ {g(} x) = gG _ {x} g  ^ {-} 1 $. 
If there is only one orbit in  $  X $, 
then  $  X $
is a homogeneous space of the group  $  G $
and  $  G $
is also said to act transitively on  $  X $. 
If  $  G $
is a [[Topological group|topological group]],  $  X $
is a topological space and the action is continuous, then  $  X/G $
is usually given the topology in which a set  $  U \subset  X/G $
is open in  $  X/G $
if and only if the set  $  \pi _ {X,G}  ^ {-} 1 ( U) $
is open in  $  X $.

1) Let  $  G $
be the group of rotations of a plane  $  X $
around a fixed point  $  a $. 
Then the orbits are all circles with centre at  $  a $(
including the point  $  a $
itself).

2) Let  $  G $
be the group of all non-singular linear transformations of a finite-dimensional real vector space  $  V $, 
let  $  X $
be the set of all symmetric bilinear forms on  $  V $, 
and let the action of  $  G $
on  $  X $
be defined by

$$ 
( gf  )( u, v)  =  f( g  ^ {-} 1 ( u), g  ^ {-} 1 ( v)) \  \textrm{ for }  \textrm{ any }
 u , v \in V.
$$

Then an orbit of  $  G $
on  $  X $
is the set of forms which have a fixed rank and signature.

Let  $  G $
be a real [[Lie group|Lie group]] acting smoothly on a [[Differentiable manifold|differentiable manifold]]  $  X $(
see [[Lie transformation group|Lie transformation group]]). For any point  $  x \in X $, 
the orbit  $  G( x) $
is an immersed submanifold in  $  X $, 
diffeomorphic to  $  G/G _ {x} $(
the diffeomorphism is induced by the mapping  $  g \mapsto g( x) $, 
$  g \in G $). 
This submanifold is not necessarily closed in  $  X $(
i.e., not necessarily imbedded). A classical example is the "winding of a toruswinding of a torus" , i.e. an orbit of the action of the additive group  $  \mathbf R $
on the torus

$$ 
T  ^ {2}  =  \{ {( z _ {1} , z _ {2} ) } : {z _ {i} \in \mathbf C ,\ 
| z _ {i} | = 1 , i = 1 , 2 } \}
$$

defined by the formula

$$ 
t( z _ {1} , z _ {2} )  =  ( e  ^ {it} z _ {1} , e ^ {i \alpha t } z _ {2} ),\ \ 
t \in \mathbf R ,
$$

where  $  \alpha $
is an irrational real number; the closure of its orbit coincides with  $  T  ^ {2} $. 
If  $  G $
is compact, then all orbits are imbedded submanifolds.

If  $  G $
is an [[Algebraic group|algebraic group]] and  $  X $
is an [[Algebraic variety|algebraic variety]] over an algebraically closed field  $  k $, 
with regular action (see [[Algebraic group of transformations|Algebraic group of transformations]]), then any orbit  $  G( x) $
is a smooth algebraic variety, open in its closure  $  \overline{ {G( x) }}\; $(
in the Zariski topology), while  $  \overline{ {G( x) }}\; $
always contains a closed orbit of the group  $  G $(
see [[#References|[5]]]). In this case the morphism  $  G \rightarrow G( x) $, 
$  g \mapsto g( x) $, 
induces an isomorphism of the algebraic varieties  $  G/G _ {x} $
and  $  G( x) $
if and only if it is separable (this condition is always fulfilled if  $  k $
is a field of characteristic zero, cf. [[Separable mapping|Separable mapping]]). The orbits of maximal dimension form an open set in  $  X $.

The description of the structure of an orbit for a given action usually reduces to giving in each orbit a unique representative  $  x $, 
the description of the stabilizer  $  G _ {x} $
and the description of a suitable class of functions which are constant on the orbit (invariants) and which separate various orbits; these functions enable one to describe the location of the orbits in  $  X $(
orbits are intersections of their level sets). This program is usually called the problem of orbit decomposition. Many classification problems can be reduced to this problem. Thus, Example 2) is a classification problem of bilinear symmetric forms up to equivalence; the invariants in this case — the rank and signature — are "discrete" , while the stabilizer  $  G _ {f} $, 
where  $  f $
is non-degenerate, is the corresponding orthogonal group. The classical theory of the Jordan form of matrices (as well as the theory of other normal forms of matrices, cf. [[Normal form|Normal form]]) can also be incorporated in this scheme: The Jordan form is a canonical representing element (defined, admittedly, up to the order of Jordan blocks) in the orbit of the [[General linear group|general linear group]]  $   \mathop{\rm GL} _ {n} ( \mathbf C ) $
on the space of all complex  $  ( n \times n) $-
matrices, for the conjugation action  $  Y \mapsto AYA  ^ {-} 1 $; 
the coefficients of the characteristic polynomial of a matrix  $  Y $
are important invariants (which, however, do not separate any two orbits). The idea of considering equivalent objects as orbits of a group is actively used in various classification problems, for example, in algebraic [[Moduli theory|moduli theory]] (see [[#References|[10]]]).

If  $  G $
and  $  X $
are finite, then the [[Burnside Lemma]] holds:

$$ 
| X/G |  =  
\frac{1}{| G | }
 \sum _ {g \in G } |  \mathop{\rm Fix}  g |,
$$

where  $  | Y | $
is the number of elements of the set  $  Y $, 
and

$$ 
 \mathop{\rm Fix}  g  =  \{ {x \in X } : {g( x) = x } \}
.
$$

If  $  G $
is a compact Lie group acting smoothly on a connected smooth manifold  $  X $, 
then the orbit structure of  $  X $
is locally finite, i.e. for any point  $  x \in X $
there is a neighbourhood  $  U $
such that the number of conjugacy classes of different stabilizers  $  G _ {y} $, 
$  y \in U $, 
is finite. In particular, if  $  X $
is compact, then the number of different conjugacy classes of stabilizers  $  G _ {y} $, 
$  y \in X $, 
is finite. For any subgroup  $  H $
in  $  G $, 
each of the sets

$$ 
X _ {(} H)  =  \{ {x \in X } : {G _ {x}  \textrm{ is }  \textrm{ conjugate }  roman ^ {  }  H
  \mathop{\rm in}  G } \}
$$

is the intersection of an open and a closed  $  G $-
invariant subset in  $  X $. 
Investigation of  $  X _ {(} H) $
in this case leads to the classification of actions (see [[#References|[1]]]).

Analogues of these results have been obtained in the geometric theory of invariants (cf. [[Invariants, theory of|Invariants, theory of]]) (see [[#References|[3]]]). Let  $  G $
be a reductive algebraic group acting regularly on an affine algebraic variety  $  X $(
the base field  $  k $
is algebraically closed and has characteristic zero). The closure of any orbit contains a unique closed orbit. There exists a partition of  $  X $
into a finite union of locally closed invariant non-intersecting subsets,  $  X = \cup _  \alpha  X _  \alpha  $, 
such that: a) if  $  x, y \in X _  \alpha  $
and  $  G( x) $
is closed, then the stabilizer  $  G _ {y} $
is conjugate in  $  G $
to a subgroup in  $  G _ {x} $, 
while if  $  G( y) $
is also closed, then  $  G _ {y} $
is conjugate to  $  G _ {x} $; 
b) if  $  x \in X _  \alpha  $, 
$  y \in X _  \beta  $, 
$  \alpha \neq \beta $, 
and  $  G( x) $
and  $  G( y) $
are closed, then  $  G _ {x} $
and  $  G _ {y} $
are not conjugate in  $  G $. 
If  $  X $
is a smooth algebraic variety (for example, in the important case of a rational linear representation of  $  G $
in a vector space  $  V = X $), 
then there is a non-empty open subset  $  \Omega $
in  $  X $
such that  $  G _ {x} $
and  $  G _ {y} $
are conjugate in  $  G $
for any  $  x, y \in \Omega $. 
The latter result is an assertion about a property of points in general position in  $  X $, 
i.e. points of a non-empty open subset; there are also a number of other assertions of this type. For example, for a rational linear representation of a semi-simple group  $  G $
in a vector space  $  V $, 
the orbits of the points in general position are closed if and only if their stabilizers are reductive (see [[#References|[7]]]); when  $  G $
is irreducible, an explicit expression of the stabilizers of the points in general position has been found (see [[#References|[8]]], [[#References|[9]]]). The study of orbit closures is important in this context. So, the set of  $  x \in V $
the closure of whose orbits contains the element  $  O $
of  $  V $
coincides with the variety of the zeros of all non-constant invariant polynomials on  $  V $; 
in many cases, and especially in the applications of the theory of invariants to the theory of moduli, this variety plays a vital part (see [[#References|[10]]]). Any two different closed orbits can be separated by invariant polynomials. The orbit  $  G( x) $
is closed if and only if the orbit of the point  $  x $
relative to the normalizer of  $  G( x) $
in  $  G $
is closed (see [[#References|[4]]]). The presence of non-closed orbits is connected with properties of  $  G $; 
if  $  G $
is unipotent (and  $  X $
is affine), then any orbit is closed (see [[#References|[6]]]). One aspect of the theory of invariants concerns the study of orbit decompositions of different concrete actions (especially linear representations). One of these — the adjoint representation of a reductive group  $  G $— 
has been studied in detail (see, for example, [[#References|[11]]]). This study is connected with the theory of representations of the group  $  G $; 
see [[Orbit method|Orbit method]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Palais, "The classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070010/o070010153.png" />-spaces" , Amer. Math. Soc. (1960) {{MR|177401}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9 {{MR|0256912}} {{MR|0256911}} {{MR|0250930}} {{ZBL|0196.27202}} {{ZBL|0193.28103}} {{ZBL|0182.57702}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Luna, "Slices étales" ''Bull. Soc. Math. France.'' , '''33''' (1973) pp. 81–105 {{MR|0342523}} {{ZBL|0286.14014}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Luna, "Adhérence d'orbite et invariants" ''Invent. Math.'' , '''29''' : 3 (1975) pp. 231–238 {{MR|0376704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R. Steinberg, "Conjugacy classes in algebraic groups" , ''Lect. notes in math.'' , '''366''' , Springer (1974) {{MR|0352279}} {{ZBL|0281.20037}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.L. Popov, "Stability criteria for the action of a semisimple group on a factorial manifold" ''Math. USSR Izv.'' , '''4''' (1970) pp. 527–535 ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''34''' (1970) pp. 523–531 {{MR|}} {{ZBL|0261.14011}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.M. Popov, "Irreducible semisimple linear Lie groups with finite stationary subgroups of general position" ''Funct. Anal. Appl.'' , '''12''' : 2 (1978) pp. 154–155 ''Funkts. Anal. i Prilozhen.'' , '''12''' : 2 (1978) pp. 91–92 {{MR|0498913}} {{ZBL|0404.22018}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.G. Elashvili, "Stationary subalgebras of points of the common state for irreducible Lie groups" ''Funct. Anal. Appl.'' , '''6''' : 2 (1972) pp. 139–148 ''Funkts. Anal. i Prilozhen.'' , '''6''' : 2 (1972) pp. 65–78 {{MR|}} {{ZBL|0252.22016}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> D. Mumford, J. Fogarty, "Geometric invariant theory" , Springer (1982) {{MR|0719371}} {{ZBL|0504.14008}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> B. Kostant, "Lie group representations on polynomial rings" ''Amer. J. Math.'' , '''85''' : 3 (1963) pp. 327–404 {{MR|0158024}} {{MR|0150240}} {{ZBL|0248.20056}} {{ZBL|0124.26802}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.L. Popov, "Modern developments in invariant theory" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , Amer. Math. Soc. (1987) pp. 394–406 {{MR|0934239}} {{ZBL|0679.14024}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Kraft, "Geometrische Methoden in der Invariantentheorie" , Vieweg (1984) {{MR|0768181}} {{ZBL|0569.14003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Kraft (ed.) P. Slodowy (ed.) T.A. Springer (ed.) , ''Algebraische Transformationsgruppen und Invariantentheorie'' , ''DMV Sem.'' , '''13''' , Birkhäuser (1989) {{MR|1044582}} {{ZBL|0682.00008}} </TD></TR></table>