characterization of signature of a permutation
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characterization of signature of a permutation
The signature of a permutation is well-defined, as proved in the parent article. This note characterizes odd permutations.
Theorem 1.
A permutation is odd if and only if the number of even-order cycles in its cycle type is odd.
Thus, for example, this theorem asserts that is an even permutation, since it has zero even-order cycles, while is odd, since it has precisely one even-order cycle.
Proof.
Note that the function taking a permutation to its signature is a homomorphism from , and we thus get the following multiplication rules for even and odd permutations:
| (even)(even) = (odd)(odd) = (even) |
| (even)(odd) = (odd)(even) = (odd) |
Note that we can represent a single cycle as a product of transpositions:
and that therefore an even-length cycle is odd (since it is equivalent to an odd number of transpositions) while an odd-length cycle is even.
By the multiplication rules above, then, a given permutation is odd if and only if the product of the signs of its cycles is odd, which happens if and only if there are an odd number of cycles whose sign is odd, which happens if and only if there are an odd number of cycles of even length. ∎
Mathematics Subject Classification
03-00 General reference works (handbooks, dictionaries, bibliographies, etc.)05A05 Permutations, words, matrices
20B99 None of the above, but in MSC2010 section 20Bxx
