Transitive action on two sets!

Question

Suppose $G$ is a finite group and G acts transitively on sets $X$ and $Y$. Let $a$ and $b$ belongs to $X$ and $Y$ respectively and $G_{a}$ be stabilizer of $a$ in $X$ and $G_{b}$ be stabilizers of $b$ in $Y$.Let $G$$=$$G_{a}$.$G_{b}$ then show that $G$$=$$G_{x}$.$G_{y}$ $\forall x \in X$, $\forall y \in Y$

Edit:$G_{x}$ denotes stabilizer of $x$ in $X$ and $G_{y}$ denotes stabilizers of $y$ in $Y$

Answer 1

Here is a hint on how to break this problem up into smaller pieces. Try to prove the following :

1. If $G$ acts transitively on a set, then any two point stabilizers are conjugate.

2. If $G = HK$ is the product of two subgroups, then $G = H^x K^y$ for any $x,y \in G$.

To prove the second fact, it helps to write $x$ and $y$ in the form $hk$, and use

3. If $G = HK$, then $HK = KH$.