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: $S_n$ denotes the symmetric group over a set of $n$ elements. It's the group of all posible permutations, so you have to find $Z(S_1),Z(S_2),...$ so you have to find the permutations that commute with every other permutations. That is the definition of $Z(G)$:
$$Z(G)=\lbrace g\in G,ga=ag\;\;\forall a\in G\rbrace$$
So it's all the elements of the group that commute with ALL members of the group. It's a generalization of the centralizer of a subgroup: if you have $H\subset G$, being $G$ a group and $H$ a subgroup, then the centralizer of $H$ in $G$ is:
$$C_G(H)=\lbrace g\in G, gh=hg\;\;\forall h\in H\rbrace$$
So the center of a group, $Z(G)$ is the centralizer of $G$ on $G$: $C_G(G)$
There are some trivial cases for little values of $n$, for example for $n=2$ the group is abelian so $Z(S_2)=S_2$. Remember the order of $S_n$: $|S_n|=n!$
