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 $\mathcal R$ 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 $G$.