I am reading about doubly transitive group actions. I am trying to prove that if $G$ acts doubly transitively on $X$, then $G_x$ is maximal for all $x\in X$ ($G_x$ is the stabilizer of $x$).
A part of the argument goes like this: Let $G_x\subsetneq H\subsetneq G$ for some subgroup $H$ of $G$. Let $Y=\lbrace hx \text{ }\mid\text{ }h\in H\rbrace$. Then $Y\neq X$.
My question is why $Y\neq X$ ?
Pick $g\in G\setminus H$. Then $gx\notin Y$. Indeed suppose $gx=hx$ for some $h\in H$. Then $h^{-1}gx=x$, so $h^{-1}g\in G_x\subseteq H$, so $g=h(h^{-1}g)\in H$, a contradiction.