S_n has Trivial Center for n greater than 3

We will prove that for $n\geq 3$ , $Z(S_n)=1$ .

Proof:

Clearly, $1\in Z(S_n)$ . Let $1\neq \sigma\in S_n$ . There exists $a, b$ distinct elements of $\{1, 2, \dots , n\}$ such that $\sigma(a)=b$ .

Consider the transposition $\tau =(b\ c)$ , where $c$ is distinct from $a, b$ . (Since $n\geq 3$ , such a $c$ exists.)

Then, $\tau\sigma (a)=\tau (b)=c$

$\sigma\tau (a)=\sigma (a)=b$

$\therefore \tau\sigma\neq\sigma\tau$

$\therefore \sigma \notin Z(S_n)$ .

Therefore, $Z(S_n)=1$

Author: mathtuition88

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