Let $G$ be a group and $A$ & $B$ be two sets s.t. $G$ acts transitively on each of $A$ & $B$. Choose some $\alpha$ and $\beta$ in $A$ & $B$ respectively then prove that if $G=G_\alpha G_\beta$ then $G=G_x G_y$ for all $x$ $\in A$ & $y$ $\in B$ where $G_x$ is stabilizer of $x$ in $G$ and similarly $G_\alpha$ be the stabilizer of $\alpha$ and $G_y$ is the stabilizer of $y$.
I know we have to choose some elements and apply some manipulative trick. it is hardly 2 liner but just can't think of right elements. help!
$G = G_\alpha G_\beta$ is equivalent to $G_\alpha$ acting transitively on $B$, which implies that any conjugate of $G_\alpha$ in $G$ is transitive on $B$. But the conjugates of $G_\alpha$ are the stabilizers $G_x$, so we get $G = G_x G_\beta$ for all $x \in A$. Then, conjugating this equation by elements of $G$ gives $G = G_x G_y$ for all $x \in A$, $Y \in B$.