Math 2.0

Group theory and generalizations Proposition

The symmetric group is generated by transpositions

Builds on Decomposition of a permutation into transpositionsSubgroup generated by a subset

Statement

Let n≥2n \geq 2. The symmetric group SnS_n is generated by its transpositions.

Remarks

This is a reformulation of the decomposition into transpositions. The adjacent transpositions (1 2),(2 3),…,(n−1 n)(1\ 2), (2\ 3), \dots, (n-1\ n) already generate SnS_n. The two elements (1 2)(1\ 2) and (1 2 … n)(1\ 2\ \dots\ n) also generate SnS_n.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Cycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementTransposition, definition, no evidence levelTranspositiondefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionDecomposition of a permutation into transpositions, statement, no evidence levelDecomposition of a pe…statementThe symmetric group is generated by transpositions, statement, no evidence levelThe symmetric group i…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. Subgroup definition
  4. Symmetric group definition
  5. Subgroup criterion statement
  6. Disjoint cycles commute statement
  7. Transposition definition
  8. Intersection of subgroups is a subgroup statement
  9. Decomposition of a permutation into disjoint cycles statement
  10. Subgroup generated by a subset definition
  11. Decomposition of a permutation into transpositions statement
  12. The symmetric group is generated by transpositions statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Symmetric_group, Generators and relations
  2. Web https://en.wikipedia.org/wiki/Cyclic_permutation, Properties
About this page
Tag0AQNX7G443
URIm2://0AQNX7G443
Kindstatement, proposition
WritingW0 stub
MSC20B30
StatusThis copy of the site has no build data. Run m2 status 0AQNX7G443.
UsesDecomposition of a permutation into transpositions, Subgroup generated by a subset

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