# Math Help - Permutation

1. ## Permutation

Show that if pi and sigma are permutations such that (pi*sigma)^2 =(pi^2)(sigma^2) then (pi*sigma)=(sigma*pi)

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

$(ab)^2 = a^2b^2$
$abab=aabb$
$ab=ba$

3. Originally Posted by r7iris
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