Let $n\geqslant 3$. Show that the unique element $\sigma$ of $S_n$ that satisfies $\sigma\gamma=\gamma\sigma$ for all of $\gamma\in S_n$ is the identity(id.)
Prepostion: If $\alpha,\beta$ are disjoint, they commute.
If we have $\alpha\in S_n$, then $id\circ\alpha=\alpha\circ id=\alpha$
I am not understanding what is asked. If the permutation are disjoint or if they are elevated to a certain power(ex:$\alpha^2=\alpha\circ\alpha)$, they commute. How can id(identity) be the only element that assures $\sigma\gamma=\gamma\sigma$?