Exercise · ZW-SIGN-OF-A-PERMUTATION-010 · ★★★ Advanced
Can \((1\ 2\ 3\ 4)\) be written as a product of two transpositions?
Choose one option.
Show answer
- C. No, because it is odd and that product is even
Explanation
No. A product of two transpositions is even, and \((1\ 2\ 3\ 4)\) is odd (a product of 3 transpositions); the parity does not depend on the factorization.
Review the concept: Sign of a permutation
The sign \(\operatorname{sgn}(\sigma)\) of a permutation \(\sigma\) is \(+1\) or \(-1\) according to whether \(\sigma\) is a product of an even or odd number \(k\) of transpositions.
\[\operatorname{sgn}(\sigma)=(-1)^k\]
More practice problems: sign of a permutation
- ZW-SIGN-OF-A-PERMUTATION-001 Compute the sign of the cycle (1\ 2\ 3).
- ZW-SIGN-OF-A-PERMUTATION-002 Compute the sign of the cycle (1\ 2\ 3\ 4).
- ZW-SIGN-OF-A-PERMUTATION-003 What is the sign of (1\ 2)(3\ 4\ 5)?
- ZW-SIGN-OF-A-PERMUTATION-004 The permutation σ∈ S_5 sends 1mapsto3, 2mapsto1, 3mapsto2, 4mapsto5, 5mapsto4. What is its sign?
- ZW-SIGN-OF-A-PERMUTATION-005 For σ∈ S_4 with list [2,4,1,3] (that is, 1mapsto2, 2mapsto4, 3mapsto1, 4mapsto3), the sign is (-1)^…
- ZW-SIGN-OF-A-PERMUTATION-006 What is the sign of a cycle of length 7?
- ZW-SIGN-OF-A-PERMUTATION-007 If σ and τ are odd permutations in S_n, what is the sign of στ?
- ZW-SIGN-OF-A-PERMUTATION-008 How many even permutations does S_4 have?