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.

IMHO, the simplest proof (one of the answers in the reference given upwards, but not selected as "the" answer) is by using the signature function. — Jean Marie, Aug 29 '17 at 17:24

score 3 · Accepted Answer · answered Aug 29 '17 at 17:02

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

answered Aug 29 '17 at 17:02

Angina Seng

161,540

could you elaborate little more how it will prove the original statement – Aug 29 '17 at 17:07
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@babai The identity permutation has $n$ cycles. – Angina Seng Aug 29 '17 at 17:07

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

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