Orbit

Let a group action of a group $G$ on a set $X$ be given, written $g\cdot x$. For $x\in X$, the orbit of $x$ is the subset

Orbits are precisely the equivalence classes of the relation “$x$ and $y$ lie in the same orbit,” and hence the set of all orbits forms a partition of $X$. The size of an orbit is controlled by its stabilizer via the orbit-stabilizer theorem .

Examples:

• For the natural action of $S_3$ on $\{1,2,3\}$, every point has orbit $\{1,2,3\}$, so the action is transitive .
• For the action of $\mathbb{Z}$ on $\mathbb{R}$ by translations $n\cdot x = x+n$, the orbit of $x$ is $x+\mathbb{Z}$.
• For the conjugation action of a group on itself, the orbits are the conjugacy classes .