Equivalence of Definitions of Even Permutation
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Theorem
The following definitions of the concept of Even Permutation are equivalent:
Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\rho \in S_n$ be a permutation in $S_n$.
Definition $1$
$\rho$ is an even permutation if and only if $\rho$ is equivalent to an even number of transpositions.
Definition $2$
$\rho$ is an even permutation if and only if:
- $\map \sgn \rho = 1$
where $\sgn$ denotes the sign function.
Proof
The sign of $\rho$ is defined as:
- $\map \sgn \rho = \begin {cases}
1 & : \text {$k$ even} \\ -1 & : \text {$k$ odd} \\ \end {cases}$
The result follows.
$\blacksquare$