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Math Help - group permutations 3

  1. #1
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    group permutations 3

    Prove that f and g^-1fg, for any f,g in Sn, are of the same parity.

    Please show steps. Thanks!
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  2. #2
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    Quote Originally Posted by mpryal View Post
    Prove that f and g^-1fg, for any f,g in Sn, are of the same parity.

    Please show steps. Thanks!
    Notice g^{-1}(1,2)g = (g^{-1}(1),g^{-1}(2)).

    Write f=t_1...t_m for transpositions t. Then g^{-1}fg = (g^{-1}t_1g)(g^{-1}t_2g)...(g^{-1}t_mg). Each g^{-1}t_jg is a transposition by the above sentence. Thus, f,g^{-1}fg both have m components in its transpotitions so they have the same parity.
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