Math 2.0

Group theory and generalizations Proposition

Sign of a cycle

Builds on Sign of a permutationDecomposition of a permutation into transpositions

Statement

Let k≥1k \geq 1 and let σ\sigma be a kk-cycle in SnS_n. Then sgn⁡(σ)=(−1)k−1\operatorname{sgn}(\sigma) = (-1)^{k-1}.

Remarks

A kk-cycle is a product of k−1k - 1 transpositions, which gives the formula. Thus a cycle is an even permutation exactly when its length is odd. With the disjoint cycle decomposition, the sign of σ∈Sn\sigma \in S_n is (−1)n−c(-1)^{n - c}, where cc is the number of cycles, with the fixed points counted as 11-cycles.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Cycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptSymmetric group, definition, no evidence levelSymmetric groupdefinitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementTransposition, definition, no evidence levelTranspositiondefinitionDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementDecomposition of a permutation into transpositions, statement, no evidence levelDecomposition of a pe…statementSign of a permutation, definition, no evidence levelSign of a permutationdefinitionSign of a cycle, statement, no evidence levelSign of a cyclestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cycle notation concept
  2. Group concept
  3. Symmetric group definition
  4. Disjoint cycles commute statement
  5. Transposition definition
  6. Decomposition of a permutation into disjoint cycles statement
  7. Decomposition of a permutation into transpositions statement
  8. Sign of a permutation definition
  9. Sign of a cycle statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, Properties
  2. Web https://en.wikipedia.org/wiki/Cyclic_permutation, Basic properties
About this page
Tag0V0E04DWX2
URIm2://0V0E04DWX2
Kindstatement, proposition
WritingW0 stub
MSC20B30
StatusThis copy of the site has no build data. Run m2 status 0V0E04DWX2.
UsesSign of a permutation, Decomposition of a permutation into transpositions

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