# How to Write Permutation as the Product of Transpositions?

I have the following problem

I have asked the question here Writing a Permutation as a product of Disjoint Cycles and got the answer.Now how can i write this same permutation as a product of Transpositions.I know how to express it as a product of disjoint cycles and i know a transposition is a cycle with 2 elements.

## 1 Answer

If you decompose into cycles first, all you need to do is express each cycle as a product of transpositions. There are various ways to do this, for example $$(1\,2\,3\,4\,\ldots\,n) = (1\,n)\cdots(1\,4)(1\,3)(1\,2)$$ or $$(1\,2\,3\,4\,\ldots\,n) = (1\,2)(2\,3)(3\,4)\cdots(n{-}1\;n)$$

• Hmm... I expressed the permutation as the product of disjoint cycles (1532)(4) then what should i do? – techno Oct 8 '15 at 11:27
• @techno: As I wrote: "all you need to do is express each cycle as a product of permutations". – hmakholm left over Monica Oct 8 '15 at 11:27
• @techno represent each cycle as a product of transpositions and put them together – Omnomnomnom Oct 8 '15 at 11:27
• @Omnomnomnom well i dont know how to represent each cycle as a transposition.Noob here :) – techno Oct 8 '15 at 11:28
• @techno: I suggest that you read my answer, immediately above this comment thread. It will tell you how to represent a cycle as a product of transpositions. – hmakholm left over Monica Oct 8 '15 at 11:29