Theorem 1: The product of disjoint cycles is commutative.
Proof: Let and be any two disjoint cycles, i.e. there is no element common in two when they are expressed in one row notation. Therefore, the elements permuted by are invariant under and the elements permuted by are invariant under .
Hence the product of disjoint cycles is commutative.
Theorem 2: Every permutation can be expressed as a composite of disjoint cycles.
Proof: Let the given permutation be denoted by the usual two row symbol within a bracket. Let be any element in the first row and the element in the second row exactly beneath , i.e. . Similarly, let . Continuing this process, an element 1 may be found in the upper row such that its image is . Then is one circular permutation. If there are additional elements , etc., in the original permutation , follow the above process to obtain another cycle . Even now, if some element or elements are left in the original permutation this procedure can be repeated to the extent that all the elements of are exhausted. In this way the original permutation can be put as the product of disjoint cycles.
Theorem 3: Every permutation can be expressed as a product of transpositions.
Proof: To prove the above result, we shall first show that every cycle can be expressed as a composite of transpositions. Let us consider a cycle then
We have already proved that every permutation can be expressed as a composition of disjoint cycles. Therefore in the light of the two results stated above, every permutation can be expressed as a product of transpositions.