Let $\displaystyle \alpha$ be a map transpositions to transpositions. If $\displaystyle x \in S_n$ then $\displaystyle i_x \in Aut(S_n)$ maps transpositions to transpositions.

Show that the set of $\displaystyle \alpha \in Aut(S_n)$ which maps transpositions to transpositions is a subgroup of $\displaystyle Aut(S_n)$.

Help this to prove.