I have the following problem: Let $G$ be a finite group with transitive group action on a set $X$ (#$X > 1$). Prove: there exists an element $g \in G$ such that $gx \neq x$ for all $x \in X$.
It would be nice if anyone could help me getting started. Usually with these problems, I have trouble understanding what information I have given. I know that transitive action is: $Gx = X$ for some $x \in X$. But how can I use this definition?
Regards.