Sign of Composition of Permutations
Theorem
Let $n \in \N$ be a natural number.
Let $N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$.
Let $S_n$ denote the set of permutations on $N_n$.
Let $\map \sgn \pi$ denote the sign of $\pi$ of a permutation $\pi$ of $N_n$.
Let $\pi_1, \pi_2 \in S_n$.
Then:
- $\map \sgn {\pi_1} \, \map \sgn {\pi_2} = \map \sgn {\pi_1 \circ \pi_2}$
where $\pi_1 \circ \pi_2$ denotes the composite of $\pi_1$ and $\pi_2$.
Proof
From Sign of Permutation on n Letters is Well-Defined, it is established that the sign each of $\pi_1$, $\pi_2$ and $\pi_1 \circ \pi_2$ is either $+1$ and $-1$.
By Existence and Uniqueness of Cycle Decomposition, each of $\pi_1$ and $\pi_2$ has a unique cycle decomposition.
Thus each of $\pi_1$ and $\pi_2$ can be expressed as the composite of $p_1$ and $p_2$ transpositions respectively.
Thus $\pi_1 \circ \pi_2$ can be expressed as the composite of $p_1 + p_2$ transpositions.
From Sum of Even Integers is Even, if $p_1$ and $p_2$ are both even then $p_1 + p_2$ is even.
In this case:
- $\map \sgn {\pi_1} = 1$
- $\map \sgn {\pi_2} = 1$
- $\map \sgn {\pi_1} \, \map \sgn {\pi_2} = 1 = 1 \times 1$
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Sources
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
