Definition:Sign of Permutation on n Letters

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

• Sign of Permutation on n Letters is Well-Defined

• 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: $\S 81$