For each $n \ge 1$ compute $Z(S_n)$ Can someone please help me on how to compute $Z(S_n)$ for each $n \ge 1$?
Does this basically mean compute $Z(1), Z(2), \ldots$? Please hint me on how to compute this.
Thanks in advance.

An elementary, but very useful fact in this context is the following. 
You should know that every element of $S_{n}$ can be written as the product of disjoint cycles in an essentially unique way. Now if you take the conjugate of the cycle $(i_1, i_2, \dots, i_k)$ by $\sigma \in S_{n}$, you obtain
$$
(i_1, i_2, \dots, i_k)^\sigma = (i_1\sigma, i_2\sigma, \dots, i_n\sigma),
$$
where I am denoting by $i \sigma$ the value of $\sigma$ on $i \in \{ 1, 2, \dots , n \}$.
This can be used here, as for a group $G$ you clearly have
$$
Z(G) = \{ x \in G : y^{-1} x y = x \text{ for all $y \in G$} \}.
$$
