I have $$A_n \leq A_nC_{S_n}(\sigma) \leq S_n$$ for some $\sigma \in S_n$.
I would like to argue that $A_nC_{S_n}$ is either $A_n$ or $S_n$.
I have so far as to think if $A_nC_{S_n}$ contains an odd cycle, it generates $S_n$ but it is still somewhat incomplete.
Any help would be appreciated!