Let $G$ be a group if $G$ acts transitively on a finite set $X$ then exists one $G$-isomorphism between $X$ and $G/Stab_G(X)$
I try to use the action of $G$ in $G/Stab_G(x)$ giving by left multiplication and i define $f:x \rightarrow G/ Stab_G(x)$ define by $f(y)=gStab_G(x)$ where $g\circ y=x$ this $g$ exists because $G$ acts transitively, and i proof that $f$ is one to one and clearly is onto.
I try to elaborate one conmutative diagram but i not sure of these way is correct. Any hint or help i will be very grateful.