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Math Help - any permutation times a k-cycle times the inverse is a k-cycle

  1. #1
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    any permutation times a k-cycle times the inverse is a k-cycle

    Question: Show that if g is a k-cycle, then fgf^{-1} is also a k-cycle, for any permutation f.

    If the cycle decomposition of either f or f^{-1} was disjoint with g then they would commute and I'd be left with just g which is a k-cycle but I'm not sure that g is disjoint to either the cycle decomposition of f or f^{-1}. What am I missing?

    Thanks!
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  2. #2
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    Quote Originally Posted by temp31415 View Post
    Question: Show that if g is a k-cycle, then fgf^{-1} is also a k-cycle, for any permutation f.

    If the cycle decomposition of either f or f^{-1} was disjoint with g then they would commute and I'd be left with just g which is a k-cycle but I'm not sure that g is disjoint to either the cycle decomposition of f or f^{-1}. What am I missing?

    Thanks!
    If \sigma = (a_1,a_2,...,a_k) and \tau is any permutation then \tau \sigma \tau^{-1} = (\tau (a_1),...,\tau (a_k)).
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    If \sigma = (a_1,a_2,...,a_k) and \tau is any permutation then \tau \sigma \tau^{-1} = (\tau (a_1),...,\tau (a_k)).
    I don't think I understand. Could you explain it in a bit more detail?
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