Equivalence of Definitions of Even Permutation

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.

$\blacksquare$