Proving an Automorphism of the symmetric group $S_4$ is the identity.

Question

I have to show an automorphism of $S_4$ is the identity if it fixes the $3$-sylow groups of $S_4.$

I've shown the automorphism acts like the identity in $A_4$. I have tried to show it acts as the identity in the transpositions but I've fail in doing so.

Any hint or answer is welcome.

I know that in worst case scenario I can check all the possible cases (since they are finite and not so many) but I reckon the problem has another way of doing it.

Answer 1

Consider the normalizer of one of the $3$-Sylow groups. You can easily show this is isomorphic to $S_3$. Likewise it is easy to show that the different $3$-Sylow normalizers contain different $3$-Sylow groups.

Just as an automorphism permutes the $3$-Sylow groups, it also permutes the normalizers... but if the $3$-Sylow groups contained within the normalizers are fixed, the normalizers must also be fixed too. Now note that the intersection of two of the normalizers is generated by a transposition, so this must fixed. The result follows.