Permutation/1357642/Several properties/Exercise

We consider the permutation ${}\tau \in S_{7}$, given by the value table

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

1. Determine the cycle representation of ${}\tau$, and the range of action.
2. Compute ${}\tau ^{3}$ and the order of ${}\tau ^{3}$.
3. Determine the inversions of ${}\tau$ and the sign of ${}\tau$.
4. Express ${}\tau$ as a product of transpositions, and determine again the sign of ${}\tau$

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