Finite permutation/Cycle length/Elements/Exercise

We consider the permutation ${}F$ on

${}M=\{1,2,\ldots ,8\}\,$

given by

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

1. Establish a value table for ${}F^{2}=F\circ F$.
2. Establish a value table for ${}F^{3}=F\circ F\circ F$.
3. Show that all iterated compositions ${}F^{n}$ are bijective.
4. Determine, for every ${}x\in M$, the minimal ${}n\in \mathbb {N} _{+}$ such that

   ${}F^{n}(x)=x\,$

   holds.

5. Determine the minimal ${}n\in \mathbb {N} _{+}$ such that

   ${}F^{n}(x)=x\,$

   holds for all ${}x\in M$.

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