Definition:Parity of Permutation
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Definition
Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\rho \in S_n$, that is, let $\rho$ be a permutation of $S_n$.
The parity of $\rho$ is defined as follows:
Even Permutation
$\rho$ is an even permutation if and only if:
- $\map \sgn \rho = 1$
where $\sgn$ denotes the sign function.
Odd Permutation
$\rho$ is an odd permutation if and only if:
- $\map \sgn \rho = -1$
where $\map \sgn \rho$ denotes the sign of $\rho$.
Also defined as
Some sources define the parity of a permutation as $\mathsf{Pr} \infty \mathsf{fWiki}$ defines its sign: that is, as $1$ and $-1$.
Also see
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