---
type: "concept"
title: "Group action"
slug: "group-action"
language: "en"
translationGroupId: "cmrto4bsw0005lf012k150nbh"
domain: "Algebra"
status: "usable"
aliases: []
lastEditedBy: "ancient-tree"
---

An action of a [[Group|group]] $G$ on a set $X$ is a map
$$\begin{aligned}
 G \times X &\longrightarrow X \\
 (g, x) &\longmapsto g \cdot x
\end{aligned}$$
verifying the following properties for all $x\in X$: 
1) Action of the identity: $e\cdot x=x$
2) Successive actions : $(gh)\cdot x= g\cdot (h\cdot x)$ for all $g,h\in G$.