Does there exist a transitive action of $S_4$ on the set $\{1,2,3,4,5\}$ ?
I would say no, because the cardinality of our set is bigger than $4$, but I am not sure how to prove this. My suggestion would be that we can’t obtain the element $5$ with the cycles of $S_4$.
Thank you.
Assume such an action exists, then using the class equation, one has: $$5=|\{1,2,3,4,5\}|=|\mathfrak{S}_4.1|.$$ (a transitive action has only one orbit)
Besides, using orbit-stabilizateur lemma, $|\mathfrak{S}_4.1|$ is a divisor of $|\mathfrak{S}_4|=24$, a contradiction, since none of the divisors of $24$ are equal to $5$.
Reminder. Let $X$ be a finite set and $G$ be a group acting on $X$, the class equation states that: $$|X|=\sum_{[x]\in X/G}|G.x|.$$