Definition:Odd Permutation/Definition 2

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 $S_n$.

$\rho$ is an odd permutation if and only if:

$\map \sgn \rho = -1$

where $\sgn$ denotes the sign function.

Examples

Example: $321$

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

Also see

• Equivalence of Definitions of Odd Permutation

• Results about odd permutations can be found here.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image: Example $142$
• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 81$