Math 2.0

Group theory and generalizations Theorem

Sign is a homomorphism

Builds on Sign of a permutationGroup homomorphism

Statement

Let n≥2n \geq 2. The map sgn⁡ ⁣:Sn→{1,−1}\operatorname{sgn} \colon S_n \to \{1, -1\} is a surjective group homomorphism, where {1,−1}\{1, -1\} is a group under multiplication. That is, sgn⁡(στ)=sgn⁡(σ)sgn⁡(τ)\operatorname{sgn}(\sigma \tau) = \operatorname{sgn}(\sigma) \operatorname{sgn}(\tau) for all σ,τ∈Sn\sigma, \tau \in S_n.

Remarks

If σ\sigma is a product of mm transpositions and τ\tau is a product of ll transpositions, then στ\sigma \tau is a product of m+lm + l transpositions. The map is surjective because sgn⁡((1 2))=−1\operatorname{sgn}((1\ 2)) = -1. Its kernel is the alternating group AnA_n.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Cycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSymmetric 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 is a homomorphism, statement, no evidence levelSign is a homomorphismstatement
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. Group homomorphism definition
  4. Symmetric group definition
  5. Disjoint cycles commute statement
  6. Transposition definition
  7. Decomposition of a permutation into disjoint cycles statement
  8. Decomposition of a permutation into transpositions statement
  9. Sign of a permutation definition
  10. Sign is a homomorphism statement

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, Properties
About this page
Tag0JEJJCB0EX
URIm2://0JEJJCB0EX
Kindstatement, theorem
WritingW0 stub
MSC20B30
StatusThis copy of the site has no build data. Run m2 status 0JEJJCB0EX.
UsesSign of a permutation, Group homomorphism

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by