Math 2.0

Group theory and generalizations Theorem

Decomposition of a permutation into transpositions

Builds on TranspositionDecomposition of a permutation into disjoint cycles

Statement

Let n≥2n \geq 2. Each permutation in SnS_n is a product of transpositions.

Remarks

A kk-cycle is a product of k−1k - 1 transpositions, for example (a1 a2 … ak)=(a1 ak)(a1 ak−1)⋯(a1 a2)(a_1\ a_2\ \dots\ a_k) = (a_1\ a_k)(a_1\ a_{k-1}) \cdots (a_1\ a_2). With the disjoint cycle decomposition, this gives the result, and the identity is (1 2)(1 2)(1\ 2)(1\ 2). The decomposition is not unique, but the parity of the number of factors is unique.

Reading path

To understand this page from the ground up, read these 7 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…statement
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

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Symmetric_group, Transpositions, sign, and the alternating group
  2. Web https://en.wikipedia.org/wiki/Cyclic_permutation, Properties
About this page
Tag0XK6FECMW5
URIm2://0XK6FECMW5
Kindstatement, theorem
WritingW0 stub
MSC20B30
StatusThis copy of the site has no build data. Run m2 status 0XK6FECMW5.
UsesTransposition, Decomposition of a permutation into disjoint cycles

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