Definition:Sign of Permutation on n Letters
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Definition
Let $n \in \N$ be a natural number.
Let $N^*_{\le n}$ denote the set of natural numbers $\set {1, 2, \ldots, n}$.
Let $S_n$ denote the set of permutations on $N^*_{\le n}$.
Let $\pi \in S_n$ be a permutation of $N^*_{\le n}$.
Let $K$ be the cycle decomposition of $\pi$.
Let each cycle of $K$ be factored into transpositions.
Let $k$ be the total number of transpositions that compose $K$.
The sign of $\pi$ is defined as:
- $\forall \pi \in S_n: \map \sgn \pi = \begin {cases} 1 & : \text {$k$ even} \\ -1 & : \text {$k$ odd} \\ \end {cases}$
Also see
- Definition:Sign of Permutation: for the general ordered $n$-tuple of real numbers
Sources
- 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: $\S 81$