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Math Help - Disjoint cycles

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    Disjoint cycles

    Prove that two disjoint cycles commute
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    MHF Contributor Bruno J.'s Avatar
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    Let \alpha, \beta \in S_n be two disjoint cycles. Clearly the j \in \{1,2,...,n\} which are fixed by both are fixed by the product, in which ever order it is taken. Now consider those j \in \{1,2,...,n\} which are not fixed by both. Since \beta and \alpha are disjoint, for a fixed j \in \{1,2,...,n\} which is not fixed by both, exactly one of \beta or \alpha moves j. So, without loss of generality, let \alpha(j)=k, \beta(j)=j with j\neq k. Since k is part of the cycle \alpha, it must not be part of the cycle \beta because \beta and \alpha are disjoint. Then \alpha\beta(j)=\alpha(j)=k and \beta\alpha(j)=\beta(k)=k, so both \beta\alpha and \alpha\beta are the same function on \{1,2,...n\}.
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