Math 2.0

Group theory and generalizations Definition

Even and odd permutations

Builds on Sign of a permutation

Definition

Let n≥2n \geq 2 and let σ∈Sn\sigma \in S_n. The permutation σ\sigma is even when sgn⁡(σ)=1\operatorname{sgn}(\sigma) = 1, and odd when sgn⁡(σ)=−1\operatorname{sgn}(\sigma) = -1.

Remarks

The identity is even and each transposition is odd. A product of two even or two odd permutations is even, and a product of an even and an odd permutation is odd. For n≥2n \geq 2, SnS_n has n!/2n!/2 even and n!/2n!/2 odd permutations. The odd permutations do not form a subgroup, but they form a coset of AnA_n.

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 permutationdefinitionEven and odd permutations, definition, no evidence levelEven and odd permutat…definition
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. Even and odd permutations definition

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, lead section
  2. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, Properties
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Tag0771AJP4BX
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Aliasesparity-of-a-permutation
Kinddefinition
WritingW0 stub
MSC20B30
DefinesEven and odd permutations
UsesSign of a permutation

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