[[Group action]]
# Orbit

Given an [[Group action|action]] of a group $G$ on a set $\Omega$,
the **orbit**[^Bahn] $G\omega$ of a point $\omega \in \Omega$ is the set of points that $m$ may be mapped to when acted upon, i.e. #m/def/group 
$$
\begin{align*}
G\Omega = \{ g\omega : \omega \in \Omega \}
\end{align*}
$$
[^Bahn]: German _Bahn_

It follows the restriction of an action onto an orbit is [[Transitive group action|transitive]],
and the induced subgroup of $M!$ is called the **transitive constituent**.

## Properties

- [[Orbit cardinality divides the finite order of the group element]]
- [[Orbit-stabilizer theorem]]
- Since orbits partition $\Omega$, one can form an [[orbit space]] $\Omega / G$.
- Iff $G\omega = \Omega$ the action is said to be [[Transitive group action|transitive]].

## See also

- [[Group action orbital]], and the more general [[n-orbit]].
- [[Group action suborbit]]

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