S_n has Trivial Center for n greater than 3

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


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

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Author: mathtuition88

Math and Education Blog

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