Let \(G\) be a group with identity \(e\), and let \(X\) be a set. An action of \(G\) on \(X\) is a map \[ G \times X \to X, \qquad (g,x) \mapsto g\cdot x, \] such that for all \(g,h\in G\) and \(x\in X\):
\(e\cdot x = x\);
\(g\cdot (h\cdot x) = (gh)\cdot x\).
Let \(G\) be a group, let \(X\) be a set, and let \(\Aut(X)\) be the group of bijections \(X\to X\) under composition. Actions of \(G\) on \(X\) correspond bijectively to group homomorphisms \(\phi\colon G\to\Aut(X)\): an action corresponds to \(\phi(g)(x) \coloneqq g\cdot x\), and a homomorphism \(\phi\) corresponds to the action \(g\cdot x \coloneqq \phi(g)(x)\).