Let \(G\) be a group acting on a set \(X\). The action is transitive if for all \(x, y\in X\) there exists \(g\in G\) such that \(g\cdot x = y\).
For nonempty \(X\), the action is transitive if and only if \(X\) is a single orbit: \(G\cdot x=X\) for some, equivalently every, \(x\in X\).