The center of $S_n$ is trivial for $n\geq 4$

Show that the center of $S_n$ for $n\geq 4$ (and in fact $n \ge 3$) is trivial.

Solution

Let $\sigma\in Z(S_n)$ with $n\ge3$. Suppose $\sigma\ne1$. Choose $a$ with \[ \sigma(a)=b\ne a, \] and choose $c$ distinct from $a,b$. Let \[ \tau=(bc). \] Then $\tau(a)=a$, so \[ (\sigma\tau)(a)=\sigma(a)=b, \] whereas \[ (\tau\sigma)(a)=\tau(b)=c. \] Thus $\sigma\tau\ne\tau\sigma$, contradicting centrality. Hence \[ \boxed{Z(S_n)=\{1\}\qquad(n\ge3)}. \]