Sign/Transferring from ordered set to arbitrary set/Remark

Let ${}I$ be an arbitrary set with ${}n$ elements, but without an ordering, and let ${}\pi$ be a permutation on ${}I$. Then we can not talk about inversions, and the definition of sign via products of differences is not directly applicable. However, we can look at fact in order to define the sign in this slightly more general situation. For this, we write ${}\pi$ as a product of ${}r$ transpositions and define

${}\operatorname {sgn} (\pi )={\begin{cases}1\,,{\text{ in case }}r{\text{ is even}}\,,\\-1\,,{\text{ in case }}r{\text{ is odd}}\,.\end{cases}}\,$

To see that this is well-defined, we consider a bijection

$\varphi \colon I\longrightarrow \{1,\ldots ,n\}.$

The permutation ${}\pi$ on ${}I$ defines on ${}\{1,\ldots ,n\}$ the permutation ${}\pi '=\varphi \pi \varphi ^{-1}$. Let ${}\pi =\tau _{1}\cdots \tau _{r}$ be a representation as a product of ${}r$ transpositions on ${}I$. Then

${}\pi '=\varphi \pi \varphi ^{-1}=\varphi \tau _{1}\cdots \tau _{r}\varphi ^{-1}=\varphi \tau _{1}\varphi ^{-1}\varphi \tau _{2}\varphi ^{-1}\varphi \cdots \varphi ^{-1}\varphi \tau _{r}\varphi ^{-1}=\tau _{1}'\tau _{2}'\cdots \tau _{r}'\,,$

where ${}\tau _{j}'=\varphi \tau _{j}\varphi ^{-1}$. These are also transpositions, so that the parity of ${}r$ is determined by the sign of ${}\pi '$.