$G$ acts transitively on a set $X$, where $|X|=m\in\mathbb{Z}_{>0}$, if and only if there exsits $H\leq G$ with index $m$. I am trying to prove that a group $G$ acts transitively on a set $X$, where $|X|=m\in\mathbb{Z}_{>0}$, if and only if there exsits $H\leq G$ with index $m$.

Notation: $Gx$ is the orbit containg $x\in X$, $G_x$ is the stabilizer for $x$.


*

*First, suppose that $G$ acts transitively on $X$, then there is only $1$ orbit, say that it is the one containing $x\in X$ and denoted by $Gx$. Since $X=\cup_{x\in X} Gx$, then $m=|X|=|Gx|$. By the stabilizer/orbit theorem, $m=|G:G_x|$, and since $G_x \leq G$, then letting $H=G_x$, proves the first part of the theorem.

However, I am struggling with the other side; any hint would be appreciated.

Given $H \le G$ let $G$ act on the set of cosets of $H$ in $G$ by  multiplication. The number of such cosets is the index of $H$ in $G.$
