If a subgroup acts transitively on a set, then the index of the subgroup equals the index of the stabilizer? I am trying to prove the following: If a subgroup $H < G$ acts transitively on a set $X$, then $[G:H] = [G_x:H_x]$ for any $x \in X$ ($H_x$ denotes the point stabilizer of $x$ in $H$.) Any hints would be appreciated. (EDIT: I am mostly interested in the case where G and H are infinite. What if $X$ is infinite?)

For the case where $G, H$ can be infinite:

Consider $H< H
$\Rightarrow [G_x:H_x][G: G_x] = [H:H_x][G:H]$. Since $[G: G_x]=[H: H_x] = |X|$, we're done.

Note: Can I assume $[G: G_x]=[H: H_x]$ even if $X$ is not finite?