Proving that ${\rm Aut}(S_3)$ is isomorphic to $S_3$
Solution 1:
$S_3$ is generated by its elements of order $2$, and the elements of $S_3$ of order $2$ are exactly the single transpositions. So all you need to show is that if you send transpositions to transpositions (through the automorphism on $S_3$), then this is equivalent to just permuting the 3 underlying elements. And then you need that permuting the $3$ elements gives an automorphism on $S_3$ always, but this is trivial.