If is a finite simple group and is a proper, non-trivial subgroup, then divides .

What I have so far:

Define . Then is a well-defined group homomorphism with kernel . Therefore by the First Isomorphism Theorem, . However I'm stuck at this point. I think I need to prove that the image of consists of even permutations on , but I can't seem to find a way to prove that. The image must be simple as is injective, but I can't see how that helps.

Any help would be greatly appreciated. Thanks!