\[ \DeclareMathOperator{\O}{O} \DeclareMathOperator{\GL}{GL} \]

Given a group \(G\) and a set \(X\). A left action of \(G\) on \(X\) is a map \(G\times X\to X\), with the following properties:

1. \(g_1\cdot (g_2\cdot x)=(g_1g_2)\cdot x\) for all \(x\in X\) and all \(g_1,g_2\in G\).
2. \(1\cdot x=x\) for all \(x\in X\).

Similarly, a right action of \(G\) on \(X\) is defined except that composition works in reverse.

Any right action defines a left action in a canonical way, and vice versa, by the correspondence

\[ g\cdot x=x\cdot g^{-1}. \]

Examples Link to heading

• action of a topological group on itself
• action of a subgroup on its group

Matrix-valued actions Link to heading

• \(\GL(n)\) on \(\mathbb{R}^n\)
• \(\O(n)\) on \(\mathbb{R}^n\)
• \(\O(n)\) on \(\mathbb{S}^{n-1}\)

See also Link to heading

• action by homeomorphisms
• continuous action
• orbit
• transitive action
• free action