The following self-contained post provides a proof of the fact that every cycle can be expressed as a product of one or more transpositions, as stated in chapter 8, p. 83, of the book “A book of abstract algebra” by Charles C. Pinter.
To show that a cycle can be written as a product of one or more transpositions, it suffices to prove the validity of the identity , which will be done by using mathematical induction.
Assume that the identity holds for . It will be shown that it also holds for , as in .
Start by noticing that .
Due to the induction assumption, which says that a -length cycle can be decomposed into a product of transpositions, can be written as .
By combining the last two equations, the conclusion follows for .