What group-theoretic construct relates the stabilizer of two points in the same orbit under a group action?
Solution
If \(y=g\cdot x\), then \[ G_y=gG_xg^{-1}. \] Indeed, for \(h\in G_x\), \[ (ghg^{-1})\cdot y=gh\cdot x=g\cdot x=y, \] so \(gG_xg^{-1}\subseteq G_y\). Applying the same argument with \(g^{-1}\) gives the reverse inclusion. Thus stabilizers of points in the same orbit are conjugate.