Definition:Orbit (Group Theory)

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This page is about the orbit of a group action. For other uses, see Definition: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 $\mathcal R$ be the relation on $X$ defined as:

$\forall x, y \in X: x \mathrel {\mathcal R} y \iff \exists g \in G: y = g * x$

where $*$ denotes the group action.


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


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


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 $X / \mathcal R_G$ is called the set of orbits of $X$ under the action of $G$.