Prove that for n>=3, if a is in S_n and commutes with every b in S_n,
then a=(1)
let $\displaystyle (1) \neq a \in S_n.$ so there exist $\displaystyle 1 \leq i \neq j \leq n$ with $\displaystyle a(i)=j.$ since $\displaystyle n \geq 3,$ there exists $\displaystyle 1 \leq k \leq n$ such that $\displaystyle k \notin \{i,j\}.$
now let $\displaystyle b=(j \ \ k).$ then $\displaystyle (ab)(i)=a(i)=j \neq k =b(j)=(ba)(i).$ thus $\displaystyle ab \neq ba. \ \Box$