Stabilizers of points in the same orbit are conjugate

Let a group \(G\) act on a set \(X\), and let \(x_0, x_1 \in X\) lie in the same orbit, say \(x_1 = g \cdot x_0\) with \(g \in G\). Then the stabilizers satisfy \(G_{x_1} = g\, G_{x_0}\, \inverseof{g}\). In particular, if \(G\) acts transitively on \(X\), all stabilizers are conjugate.