Show that no subgroup $G$ of $S_5$ acts transitively on $X=\{1,2,3,4,5\}$ in such a way that $G_x$ is an elementary abelian 2-group for any $x\in X$
I want to solve this problem. I decided to use proof by contradiction.
By orbit-stabilizer theorem and with transitive condition, $|X|=|Gx|=|G:G_x|$. Then $|G|=|X||Gx|=5|Gx|$.
By Lagrange theorem with above condition, $|G|$ can be $5, 10, 20, 40$. But there is no progress from here.
Help me how to solve this problem!