I'm currently trying to work through a proof that Aut$(A_n) \cong S_n$. In particular I'm looking at theorem 2.3 (on page 18) in R. Wilson's book 'The Finite Simple Groups'. (Click here for a download link provided by Springer.)
If I've understood the proof correctly, it works by showing that Aut$(A_n)$ acts as $S_n$ on the set of the $n$ point-stabilisers in $A_n$. So there exists a surjective homomorphism Aut$(A_n) \rightarrow S_n$.
My question is why is this homomorphism injective, i.e. why is the above action faithful? Stated another way, why is an automorphism of $A_n$ which fixes all of the point-stabilisers the trivial automorphism?