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Thread: Monoids and Groups

  1. #1
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    Monoids and Groups

    Can I get help on the following problem:
    Show that if n GE 3 then the center of Sn is of order 1.
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    Quote Originally Posted by thomas_donald View Post
    Can I get help on the following problem:
    Show that if n GE 3 then the center of Sn is of order 1.
    Let $\displaystyle n\geq 4$.

    Let $\displaystyle \sigma$ be non-trivial permutation. Then there exists $\displaystyle a,b\in \{1,2,...,n\}$ such that $\displaystyle \sigma(a) = b$ with $\displaystyle a\not = b$. Notice that $\displaystyle a,\sigma^{-1}(a),b$ has at most three distinct points, let $\displaystyle c$ be different from all three of these (since $\displaystyle n\geq 4$ this is possible). There exists a permutation $\displaystyle \tau$ which satisfies $\displaystyle \tau(a) = \sigma^{-1}(a),\tau(b) = c$. Thus, $\displaystyle \tau \sigma(a) = \tau (b) = c$ and $\displaystyle \sigma \tau(a) = \sigma \sigma^{-1}(a) = a$. We see that $\displaystyle \tau \sigma \not = \sigma \tau$. This shows if $\displaystyle \sigma$ is not trivial then it cannot lie in the center.
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