Finite permutation/Cycle length/Elements/Exercise

We consider the permutation $F$ on

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

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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