Transitive Action of a Finite Group

Question

If $G$ is a finite group acts transitively on a set $\mathbb{X}$. Does this imply that the set $\mathbb{X}$ is finite?

Answer 1

If $G$ acts transitively on $X$, then, given $x\in X$, for every $y\in X$ there is some $g\in G$ such that $y=g\cdot x$. This is precisely the condition for a surjection $f_x\colon G\longrightarrow X$ to exist. Therefore the cardinality of $G$ is at least equal to the one of $X$.

(Incidentally, since by group action's axioms distinct $x$'s induce distinct $f_x$'s, there are $|X|$ of such surjections.)