- Show that if \(x, y\) are in the same orbit, then their stabilizers are conjugate.
Solution
Let \(G\) act on \(X\), and suppose \(y=g\cdot x\). Then \[ G_y=gG_xg^{-1}. \] Indeed, if \(h\in G_x\), then \[ (ghg^{-1})\cdot y =ghg^{-1}g\cdot x =g(h\cdot x) =g\cdot x =y, \] so \(gG_xg^{-1}\subseteq G_y\).
Conversely, if \(k\in G_y\), then \[ (g^{-1}kg)\cdot x =g^{-1}k\cdot(gx) =g^{-1}(k\cdot y) =g^{-1}y =x, \] so \(g^{-1}kg\in G_x\), hence \(k\in gG_xg^{-1}\). Therefore \[ G_y=gG_xg^{-1}. \]