---
type: "concept"
title: "Orbit-stabiliser theorem"
slug: "orbit-stabiliser-theorem"
language: "en"
translationGroupId: "cmrtortzw000hlf01apodnbwq"
domain: "Algebra"
status: "stub"
aliases: []
lastEditedBy: "ancient-tree"
---

Let $G$ a group [[Group action|acting]] on a set $X$ and $x$ in $X$. The map
$$\begin{aligned}
\varphi: G / G_x & \longrightarrow \operatorname{Orb} (x) \\
g G_x & \longmapsto g \cdot x
\end{aligned}$$
is a [[Well-defined map|well-defined]] [[Bijective map|bijection]] from the set of [[Left coset|left cosets]] of $G_{x}$ onto the [[Orbit of a group action|orbit]] of $x$.