I've been looking into the group of automorphisms of the symmetric group $S_{n}$ for when $n > 6$. Something which is claimed frequently is that if an automorphism sends a transposition to a transposition, then it has to be an inner automorphism (and is hence used to show that Aut($S_{n}$) = $S_{n}$.
However, for $n > 6$, is there a way to constructively show that any automorphism sends a transposition to a transposition?
I've been attempting to show it using centralizers but haven't had much luck.
Thanks!