Definition:Even Permutation

Definition

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.

Examples

Example: $312$

$\tuple {3, 1, 2}$ is an even permutation of $\tuple {1, 2, 3}$.

Also see

• Equivalence of Definitions of Even Permutation

• Definition:Odd Permutation

• Results about even permutations can be found here.