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Math Help - Cycle permutation problem

  1. #1
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    Cycle permutation problem

    Hello there. I have this two cycle permutation problems to prove,

    1. \left ( a b \right )^{-1} = \left ( a b \right )

    2. (1,2,3, .... ,n)^{-1} = (n,n-1, ... ,2,1)

    Any idea?
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  2. #2
    Junior Member Nehushtan's Avatar
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    Re: Cycle permutation problem

    #1. What is (a\ b)(a\ b)?

    #2. Use #1 and induction. (Note: (1\ 2\ \cdots\ n\ n+1)=(1\ n+1)(1\ 2\ \cdots\ n).)
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  3. #3
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    Re: Cycle permutation problem

    Quote Originally Posted by Nehushtan View Post
    #1. What is (a\ b)(a\ b)?

    #2. Use #1 and induction. (Note: (1\ 2\ \cdots\ n\ n+1)=(1\ n+1)(1\ 2\ \cdots\ n).)
    1. It is decomposition into disjoint cycle?
    2. I tried to solve it like this.
    \tau = (a_{1}  a_{2} ... a_{n} )
    \tau(a_{i})=a_{i+1}
    \tau^{-1}(a_{i+1})=a_{i}
    \tau: a_{1} \mapsto a_{2} \mapsto ...\mapsto a_{n}
    \tau^{-1}: a_{n} \mapsto a_{n-1} \mapsto ...\mapsto a_{1}
    - Its obvious that tau^(-1), is just tau written in reverse.
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