Definition:Orbit (Group Theory)

Definition
Let $G$ be a group acting on a set $X$.

The orbit of an element $x \in X$ is defined as:


 * $\operatorname{Orb} \left({x}\right) := \left\{{y \in X: \exists g \in G: y = g * x}\right\}$

where $*$ denotes the group action.

That is, $\operatorname{Orb} \left({x}\right) = G * x$.

Thus the orbit of an element is all its possible destinations under the group action.

Length
The length of an orbit $\operatorname{Orb} \left({x}\right)$ is the number of elements it contains, i.e. $\left|{\operatorname{Orb} \left({x}\right)}\right|$.