Group action
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Revision 532
7/20/2026, 8:43:19 PM · Ancient Tree
Concept edited
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An action of a group $G$ on a set $X$ is a map1
An action of a [[Group|group]] $G$ on a set $X$ is a map2
$$\begin{aligned}3
G \times X &\longrightarrow X \\4
(g, x) &\longmapsto g \cdot x5
\end{aligned}$$6
verifying the following properties for all $x\in X$: 7
1) Action of the identity: $e\cdot x=x$8
2) Successive actions : $(gh)\cdot x= g\cdot (h\cdot x)$ for all $g,h\in G$.Revision 531
7/20/2026, 8:43:05 PM · Ancient Tree
Concept edited
Compare with revision 5239 changed lines
1
A group action is...1
An action of a group $G$ on a set $X$ is a map2
$$\begin{aligned}3
G \times X &\longrightarrow X \\4
(g, x) &\longmapsto g \cdot x5
\end{aligned}$$6
verifying the following properties for all $x\in X$: 7
1) Action of the identity: $e\cdot x=x$8
2) Successive actions : $(gh)\cdot x= g\cdot (h\cdot x)$ for all $g,h\in G$.Revision 523
7/20/2026, 8:19:25 PM · Ancient Tree
Concept created
A group action is...