$Tran(x,y)=\{g\in G: g\cdot x=y\}$ is a lateral coset of stabilizer $G_x$. Let $G$ a group so that act transitively on a nonempty set $X$. 1. Show that the stabilizer of two different points of $X$ are conjugated. 2. Show that $Tran(x,y)=\\{g\in G: g\cdot x=y\\}$ is a lateral coset of stabilizer $G_x$. I showed the first enunciate but not the second. Thanks for your help!

Since $G$ acts transitively on $X$, note $Tran(x, y)$ is nonempty for all $x, y \in X$. Fix $g \in Tran(x, y)$, and consider the coset $gG_{x}$ of $G_{x}$. We will show $gG_{x} = Tran(x, y)$.

Let $k \in gG_{x}$, i.e. $k = gh$ for some $h \in G_{x}$. Then $kx = (gh)x = g(hx) = gx = y$, so $k \in Tran(x, y)$. Hence, $gG_{x} \subseteq Tran(x, y)$.

Now let $k \in Tran(x, y)$. Then $kx = y = gx$, so $g^{-1}kx = x$, i.e. $g^{-1}k \in G_{x}$. Hence, $g^{-1}k = h$ for some $h \in G_{x}$, i.e. $k = gh$, so $k \in gG_{x}$. Thus, $Tran(x, y) \subseteq gG_{x}$.