C4 - theorems

every permutation is a product of disjoint cycles

\begin{matrix} [\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 1 & 2 & 9 & 8 & 5 & 3 & 6 & 4 \end{matrix}] \\ = (\begin{matrix} 1 & 7 & 3 & 2 \end{matrix}) (\begin{matrix} 4 & 9 \end{matrix}) (\begin{matrix} 5 & 8 & 6 \end{matrix}) \end{matrix}

disjoint cycles commute

let $α = {a_{1}, a_{2}, \dots a_{r}} \in S_{n}$ and $β = {b_{1}, b_{2}, \dots b_{k}} \in S_{n}$ such that $α \cap β = Φ$ , then $α β = β α$

order of permutation

let $α \in S_{n}$ ,then $| α | =$ LCM of the length of disjoint cycles
consider $(\begin{matrix} a & b & c \end{matrix}) (\begin{matrix} d & e \end{matrix}) (\begin{matrix} f \end{matrix})$ , so LCM of disjoint cycles is $6$
$α^{3 m} = ϵ$ and $β^{2 n} = ϵ$ , which happens for $m = 2$ and $n = 3$ , ie. they both need to be raised to the power of $6$