# How to write the identity permutation as a product of transpositions

The book that I'm reading states that the identity permutation is an even permutation. But it gives no example, and at this point, I'm confused. So, for example, if we have the identity permutation $\varepsilon=(1)(2)(3)(4)(5)$, how do we write its product of transpositions? I tried $(12)(34)(52)$, and so on, which is obviously incorrect. Any insight and/or example would great!

• a) empty product. b) $(12)(12)$. – Daniel Fischer Oct 10 '15 at 19:48

Suppose an identity permutation as $$(1)(2)(3)$$ , then it can be written as $$(23)\times(32)$$ by remaining the position of $$1$$ unaltered. Or in similar ways it also can be written as $$(12)\times(21)$$ or as $$(13)\times(31)$$.