Permutation/8/25371486/45286713/Product/Inversions/Sign/Exercise

We consider the two permutations given by

${}x$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$	${}7$	${}8$
${}\sigma (x)$	${}2$	${}5$	${}3$	${}7$	${}1$	${}4$	${}8$	${}6$

and

${}x$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$	${}7$	${}8$
${}\tau (x)$	${}4$	${}5$	${}2$	${}8$	${}6$	${}7$	${}1$	${}3$

Compute ${}\sigma \tau$ and ${}\tau \sigma$. Determine the number of inversions and the sign of ${}\tau$. Describe the cycle representation of ${}\sigma$ and of ${}\sigma ^{3}$. What is the order of ${}\sigma$?

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