A transitive $G$-set is the coset space of any point stabilizer.Suggest a slogan
Let a group \(G\) act transitively on a set \(X\), and let \(x \in X\) with stabilizer \(G_x\). Then \[ \begin{aligned} \Phi\colon G/G_{x} &\to X \\ gG_{x} &\mapsto g \cdot x \end{aligned} \] is a well-defined bijection from the set of left cosets of \(G_x\) to \(X\), and \(\Phi(h \cdot gG_x) = h \cdot \Phi(gG_x)\) for all \(g, h \in G\).