Exercise · ZW-SIGN-OF-A-PERMUTATION-002 · ★☆☆ Basic
Compute the sign of the cycle \((1\ 2\ 3\ 4)\).
Choose one option.
Show answer
- C. \(-1\)
Explanation
A cycle of length 4 is a product of 3 transpositions, \((1\ 4)(1\ 3)(1\ 2)\); since 3 is odd, the sign is \(-1\).
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-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?
- ZW-SIGN-OF-A-PERMUTATION-009 Compute the sign of (1\ 2)(3\ 4\ 5\ 6).
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