I found this problem, and I was wondering if I'm on the right approach. Let G be a finite group on a finiste set X with m elelements. Suppose there exist a g[tex]\in[/tex]G and x[tex]\in[/tex]X such that gx not equal to x. Suppose the order of G does not divide m!. Prove that G is not simple. Would it suffice to show that an isomorphism "f" exists from G to X? Then we just need to prove two cases about the Ker(f). We need to show that Ker(f) can't just be the identity because then it would be an infinite group being isomorphic to a finite group. If the Ker(f)=G, then some stuff. Sorry for the informality, I'm not actually sure what happens if Ker(f)=G.