Let \(G\) be a group with identity \(e\) and \(X\) a set. An action of \(G\) on \(X\) is a map \(G\times X\to X\), written \((g,x)\mapsto g\cdot x\), such that for all \(g,h\in G\) and \(x\in X\), \[ e\cdot x=x \qquad\text{and}\qquad g\cdot(h\cdot x)=(gh)\cdot x. \]
For \(x\in X\), the orbit and the stabilizer of \(x\) are \[ G\cdot x\coloneqq\theset{g\cdot x \st g\in G}, \qquad G_x\coloneqq\theset{g\in G \st g\cdot x=x}. \] The fixed-point set of the action is \(X^G\coloneqq\theset{x\in X \st g\cdot x=x\text{ for every }g\in G}\).
Let \(G\) act on \(X\).
The orbits of the action partition \(X\).
The action is transitive if and only if it has exactly one orbit, provided \(X\neq\emptyset\).
The kernel of the associated homomorphism \(G\to\Sym(X)\), \(g\mapsto(x\mapsto g\cdot x)\), is \(\bigcap_{x\in X}G_x\).
Write \(x\sim y\) if \(y=g\cdot x\) for some \(g\in G\). This relation is reflexive because \(e\cdot x=x\), symmetric because \(y=g\cdot x\) gives \(g^{-1}\cdot y=(g^{-1}g)\cdot x=x\), and transitive because \(y=g\cdot x\) and \(z=h\cdot y\) give \(z=(hg)\cdot x\). Its equivalence classes are the orbits.
The action is transitive exactly when every two points of \(X\) are related by \(\sim\), that is, when the nonempty set \(X\) is a single equivalence class.
An element \(g\) acts as the identity permutation if and only if \(g\cdot x=x\) for every \(x\in X\), that is, \(g\in G_x\) for every \(x\).