Let $X$ be a finite set and $n=|X|$. Let $G$ be $k$-transitive group on $X$ for some $k$ such that $1\leq k\leq n$. If $G$ is also primitive, does this necessarily imply that $k>1$? My book says yes. I am not sure how to prove it.
I know that $G$ being primitive means that, for all $x\in G$, $G_x$ is a maximal subgroup of $G$. How does this property together with $k$-transitivity imply that $k>1$? How should I be thinking about this?
Edit:
I did not express the problem correctly. I have to prove that $k>1$ together with $k$-transitivity of $G$ imply that $G$ is primitive.