There is an exercise: If $G$ is $k$-transitive but not $(k+1)$-transitive, is it true that $G$ is sharply $k$-transitive?
I solved this exercise "if $G$ is sharply $k$-transitive then G is not $(k+1)$-transitive". I try to prove the first exercise. But I don't have any idea how to solve it. I don't know if it's true or false. I think it's false but which counterexample?
Any kind of suggestion is appreciated. Thanks to everyone for the help.