O

Abstract

Orbit of a point x relative to a group G acting on a set X (on the left) — The set

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

The set

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

is a subgroup in G and is called the stabilizer or stationary subgroup of the point x relative to G. The mapping g↦g(x), g∈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 π _X,G : X→X/G. The stabilizers of the points from one orbit are conjugate in G, or, more precisely, G _g (x)= _g G _{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, X is a topological space and the action is continuous, then X/G is usually given the topology in which a set U⊂X/G is open in X/G if and only if the set μ _X,G ^-1(U) is open in X.