Let $(G,\cdot)$ be finite group of order $m$ and $X$ be a finite set of order $n$. Show that if $m$ is a prime number greater than $n$, then the only action of G under X that is possible to define is the trivial.
Sorry but, I couldn't get any idea of resolution.
One useful perspective is to view a group action of $G$ on a finite set $X$ has a homomorphism $\varphi:G\rightarrow S_{|X|}$, where $S_{|X|}$ is the group of permutations on $|X|$ letters (it shouldn't be too hard to show that these definitions are equivalent).
Then the claim reduces to one about the existence of a nontrivial such homomorphism. You should be able to make a claim that any nontrivial homomorphism is in fact injective, and then use something about the symmetric group $S_{|X|}$ to derive a contradiction.