Show that the identity permutation cannot be expressed as the product of an odd number of transpositions.

Question

Show that the identity permutation cannot be expressed as the product of an odd number of transpositions.

For example consider the example of $S_3,$ $\sigma_{id} = (123)(321) = (1 2) (13)(32)(31)$, which means even number of permutations.

In general let $\sigma_{id}= \sigma_1.\sigma_1\cdots \sigma_k $, I need to show if I rewrite each $\sigma_i$ as transpositions then odd number of transpositions will be there.

Thank You.

Answer 1

Prove that if $\sigma\in S_n$ is a product of $r$ transpositions, and has $s$ cycles, then $r+s+n\equiv 0\pmod 2$.