Do distinct elements of $S_n$ afford distinct inner automorphisms of $S_n$?
I believe that this is true but am wondering if there is a way to prove this just by appealing to general facts about automorphisms and/or $S_n$, rather than doing it mechanically.
The map $\gamma$ from a group $G$ to its inner automorphism group, which assigns to $g$ the automorphism of conjugation by $g$, has kernel equal to the center $Z(G)$. Thus, if $G$ is centerless, $\gamma$ is 1-1, and distinct elements of $g$ afford distinct inner automorphisms. $S_n$ is centerless if $n>2$.