Let $ G $ be a finite simple group which acts on finite set $ X $ non-trivially. The goal is that the largest prime number $p$ dividing $ \left|G\right| $ is also divide $ \left|X\right| $.
I know $G$ is embedded in $S_{X}$. But, I can't get any other things.
Hmm, I'm not sure to understand exactly the question : do you want to prove this for every non-trivial action or do you want a condition for it to be true. Because I think in general it's false (I even think that for almost every group $G$ you can find a non-trivial action for which this is false).
Take $G$ of order $n=p^km$, with $m \wedge p=1$ and $p$ larger than all the prime factors of $m$. Consider a $p$-Sylow $P$ of $G$ and the quotient $X=G/P$. $G$ act transitivly and non-trivially on $X$ by translation, however $|X|=m$ so $p$ does not divide $|X|$.
Is there a mistake there ?