Definition 5

Let G be a group, E a G-set and $x \in G$. An element $y \in E$ is conjugate to x under the operation of G if there exists an element $\alpha \in G$ such that $y = \alpha x$. The set of conjugate elements of x is called the orbit of x in E.

The relation "y is conjugate to x" is an equivalence relation. For $x = ex$; if $y = \alpha x$, then $x = \alpha^{-1} y$; if $y = \alpha x$ and $z = \beta y$, then $z = \beta \alpha x$. The orbits are the equivalence classes under this relation.

A subset X of E is stable if and only if it is saturated with respect to the relation of conjugation.

The mapping $\alpha \mapsto \alpha x$ of G into E is sometimes called the orbital mapping defined by x. It is a G-morphism of G (with the operation of G on itself by left translation) into E. The image $G . x$ of G under this mapping is the orbit of x.

G is said to operate freely on E if, for all $x \in E$, the orbital mapping defined by x is injective or also if the mapping $(g, x) \mapsto (gx, x)$ of $G \times E$ into $E \times E$ is injective.