Definition:Orbit (Group Theory)

This page is about Orbit in the context of Group Action. For other uses, see Orbit.

Definition

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

Definition 1

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.

Definition 2

Let $\RR$ be the relation on $X$ defined as:

$\forall x, y \in X: x \mathrel \RR y \iff \exists g \in G: y = g * x$

where $*$ denotes the group action.

From Group Action Induces Equivalence Relation, $\RR$ is an equivalence relation.

The orbit of $x$, denoted $\Orb x$, is the equivalence class of $x$ under $\RR$.

Length

The length of the orbit $\Orb x$ of $x$ is the number of elements of $X$ it contains:

$\size {\Orb x}$

Set of Orbits

The quotient set $X / \RR_G$ is called the set of orbits of $X$ under the action of $G$.