Definition:Orbit (Group Theory)/Definition 1

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

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


 * $\Orb x := \set {y \in X: \exists g \in G: y = g * x}$

where $*$ denotes the group action.

That is, $\Orb x = G * x$.

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

Also see

 * Group Action Induces Equivalence Relation, which demonstrates the equivalence of these definitions.