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:


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

where $$*$$ denotes the group action.

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

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