Reference for ${\rm sgn} ( \sigma \tau ) = {\rm sgn} ( \sigma ) {\rm sgn} ( \tau )$ and ${\rm sgn} ( \sigma ) ={\rm sgn} ({ \sigma}^{-1} )$

Question

Does anyone know of a group theory textbook that contains a proof of the following? I would like to refer to the proof in a paper I am writing.

Let $\sigma ,\tau \in {S_n}$. We have that $\operatorname{sgn} \left( {\sigma \tau } \right) = \operatorname{sgn} \left( \sigma \right)\operatorname{sgn} \left( \tau \right)$ and $\operatorname{sgn} \left( \sigma \right) = \operatorname{sgn} \left( {{\sigma ^{ - 1}}} \right)$.

I've only been able to find textbooks that say "from the definition it is easy to check the formulas ..."

Answer 1

You could quote John F. Humphreys' book A course in group theory.

The first identity is Proposition $9.16$ and the second one is Corollary $9.17$.