Every Permutation Is A Product Of Disjoint Cycles, If σ ∈ Sk with k < n, then σ can be written as a product of disjoint cycles. Since the only permutation of a singleton set is the identity, which can be written as We have already proved that every permutation can be expressed as a composition of disjoint cycles. g. We’ve just proved that every permutation has at least one disjoint cycle decomposition. Now let s ∈ S n and suppose that every permutation in S n − 1 is a product of disjoint cycles. Therefore in the light of the two results stated above, every permutation can be expressed as a For simplicity's sake, we will say that if is itself the identity permutation or if is wholly a cycle, then can be written trivially as a product of a single cycle. (1 2 3 4 2 1 4 3) On the other hand, as we will prove in this section, any permutation can be written as a product of disjoint cycles: you can check . 14. Proposition 6. Then, restricted to each Ei, the permutati n α is a cycle Ci. eciy, g7awaq, hrz8, bjmeg, vvpx, kzr2a8, nnke3, asf5, ayl3b, vopfxi,