Let $X$ be a transitive $G$-set, and let $x_0 \in X$. Show that $X$ is isomorphic to the $G$-set $L$ of all left cosets of $G_{x_0}$
As X is a transitive $G$-set, $X=Gx_0$, so one can define $\phi$:$Gx_0$ $\to$ $L$; $\phi(gx_0)=gG_{x_0}$, now for verifying that this function is well-defined and injective, I did the following: $gx_0=hx_0$ $\iff$ $h^{-1}gx_0=x_0$ $\iff$ $h^{-1}g$ $\in$ $G_{x_0}$ $\iff$ $gG_{x_0}=hG_{x_0}$. The surjectivity is clear but for verifying the homomorphism property, which result should I check? Is that $\forall g \in G$; $\forall x \in X$ $g\phi(x)=\phi(gx)$ ? If so, $g\phi(ax_0)$=$g(aG_{x_0})$=$gaG_{x_0}$=$\phi(gax_0)$.
