I have been working on this proof for some time taking reference form Lang. I understand that if we prove that if every normal subgroup of $A_n$ contains some 3-cycles then the proof is complete.
Now I can prove till the fact that if $\sigma \in N$ where N is a non-trivial subgroup of $A_n$, and $\sigma$ is the element such that it has the maximal number of elements left unmoved, then it can't be a product of two independent 2-cycles. I can also prove the part part (b) of Lang's proof as given below:
Now my question is, why would you stop there? I mean why would you only assume that $\sigma$ moves five elements and prove that it then can't be the element in N such that it has the maximal number of unmoved points? Why can't you assume that $\sigma$ moves 6 points or more? If someone could help me out with the last part of the proof, I'd really appreciate it. Thanks!