Theorem 1

Let G be a group. (a) Every homogeneous G-set is isomorphic to a homogeneous G-set of the form G/H where H is a subgroup of G. (b) Let H and H' be two subgroups of G. The G-sets G/H and G/H' are isomorphic if and only if H and H' are conjugate.

As a homogeneous G-set is non-empty, assertion (a) follows from Proposition 6. We show (b). Let $f : G/H \to G/H'$ be a G-set isomorphism. The subgroup H is the stabilizer of H and hence, by transport of structure, the stabilizer of an element of $G/H'$. The subgroups $H$ and $H'$ are therefore conjugate (no. 2, Proposition 2). If $H' = \alpha H \alpha^{-1}$, $H'$ is the stabilizer of the element $\alpha . H$ of $G/H$ (no. 2, Proposition 2) and hence $G/H'$ is isomorphic to $G/H$ (Proposition 6).