Parity of Permutation

Definition 1. Let $\sigma \in \SS_n$ be a permutation ($n \in \N^+$). The parity of $\sigma$ is

\[\par \sigma = (-1)^k,\]

where

\[k = \big\lvert \{ % (i, j) \in \llbra n \rrbra \times \llbra n\rrbra : i < j \,\land\, \sigma(i) > \sigma(j) % \}\big\rvert\]

is the number of inversions in $\sigma$. $\sigma$ is called an even permutation if $k$ is even and an odd permutation if $k$ is odd.

Proposition 1. For any $\sigma \in \SS_n$ ($n \geq 2$),

\[\par \sigma = \prod_{1 \leq i < j \leq n} \frac{\sigma(i) - \sigma(j)}{i - j}.\]

Proposition 2. For any $n \in \N^+$,

\[\par : \SS_n \to \{-1, 1\}\]

is a group homomorphism and we have

\[\ker \par = \AA_n,\]

the alternating group of degree $n$.