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.

Hint: I think you should start with $S_3$. Just fix an arbitrary permutation and see what it means for it to commute with all the other permutations in the group. The same idea works for all $n>3$.
