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Math Help - Permutation

  1. #1
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    Permutation

    Show that if pi and sigma are permutations such that (pi*sigma)^2 =(pi^2)(sigma^2) then (pi*sigma)=(sigma*pi)
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  2. #2
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    Quote Originally Posted by r7iris View Post
    Show that if pi and sigma are permutations such that (pi*sigma)^2 =(pi^2)(sigma^2) then (pi*sigma)=(sigma*pi)
    Just expand.

    $(ab)^2 = a^2b^2$
    $abab=aabb$
    $ab=ba$
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by r7iris View Post
    Show that if pi and sigma are permutations such that (pi*sigma)^2 =(pi^2)(sigma^2) then (pi*sigma)=(sigma*pi)
    Simple. We know the inverse permutations (on the left and right) exist, so:
    ( \pi \sigma )^2 = \pi ^2 \sigma ^2

    \pi \sigma \pi \sigma = \pi ^2 \sigma ^2

    \pi \sigma \pi \sigma \sigma ^{-1} = \pi ^2 \sigma ^2 \sigma ^{-1}

    \pi \sigma \pi = \pi ^2 \sigma

    \pi ^{-1} \pi \sigma \pi = \pi^{-1} \pi ^2 \sigma

    \sigma \pi = \pi  \sigma

    -Dan
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