Theorem 1, p. 82, in the book “A book of abstract algebra” by Charles C. Pinter states that “Every permutation is either the identity, a single cycle, or a product of disjoint cycles”. The theorem is proved in the book. After the proof, it is stated that it is easy to see that the product of cycles is unique, except for the order of factors. I have seen online proofs of this claim of uniqueness based on the concept of orbit. In what follows, an alternative proof is offered without relying on orbits.
Assume that there exists a permutation which can be written as a product of disjoint cycles in two ways. So, there are two collections, each consisting of disjoint cycles, and such that and .
. Without loss of generality, assume that . Thus, .
Let be some element in cycle . Since and the cycles in are disjoint, is permuted by . Therefore, such that .
The cycles and are disjoint from the rest of cycles in and , respectively. Thus, .
Since , there exists an element that breaks the equality arising from the successive implementation of permutation in each of the two cycles and starting from . Thereby, there exists an element in and in with . This is a contradiction, since the permutation is a function, so should be mapped to a unique element .