> [!NOTE] Theorem ($k$-cycle can be factored into $(k-1)$ $2$-cycles) > Every [[Cyclic Permutation of n Letters|k-cycle]] can be factorised into $(k-1)$ $2$-cycles (or transpositions). *Proof*. The $k$-cycle $(a_{1},a_{2},\dots,a_{k})$ has the factorisation $(a_{1},a_{k})(a_{1},a_{k-1})\dots(a_{1},a_{3})(a_{1},a_{2}).$ > [!NOTE] Corollary (Permutation is a product of transpositions) > Every [[Permutation of Finite Degree|permutation]] can be written as a product of $2$-cycles. *Proof*. Follows from [[Existence of Disjoint Cycle Decomposition for Permutations of n Letters]].