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!
[Math] How to write the identity permutation as a product of transpositions
abstract-algebrapermutations
Best Answer
A permutation is even iff it is a product of an even number of transpositions. The identity permutation can be represented as a product of zero transpositions - and zero is certainly even.