If a finite group acts transitively on a set, does its center also acts transitively? > If $G$ is a finite group acts transitively on a set $X$, does the center $Z(G)$ also acts on $X$ transitively? I don't see how this statement will be true but I can't come up with a counter example either. Any help is appreciated.

Hint: Think about the Symmetric group $S_3$ acting on $\\{1,2,3\\}$. What is the center of $S_3$? In fact, any $S_n$ for $n > 2$ acting on $\\{1,2,\dots, n\\}$ will work. Why?