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Math Help - Permutations

  1. #1
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    Permutations

    Ok, I don't know how to go about this.

    Prove that there is no permutation α such that α(12) α^-1 = (123)

    Prove that there is no permutation α such that α(12) α^-1 = (124)(567)
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  2. #2
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    Quote Originally Posted by chris27 View Post
    Ok, I don't know how to go about this.

    Prove that there is no permutation α such that α(12) α^-1 = (123)

    Prove that there is no permutation α such that α(12) α^-1 = (124)(567)
    Well (1 2 3) = (1 2)(2 3), which is an even permutation.
    So \alpha (1 2) \alpha^{-1} must be even, but this is impossible because if \alpha is an even permutation, then \alpha (1 2) is odd, so \alpha (1 2)\alpha^{-1} is odd.
    But if \alpha is odd, then \alpha (1 2) is even, so \alpha (1 2) \alpha^{-1} is odd. So the left side is never an even permutation.

    The same reasoning applies to the second problem.
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  3. #3
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    Two cycles cannot be conjugate if they have different lengths for that will imply thier orders are not the same.
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