Math 2.0

statement proposition

The symmetric group is generated by transpositions

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.

Explanations

L1 human

Each permutation is a product of swaps, so the swaps generate the whole symmetric group. You do not even need all swaps. Swaps of neighbours are enough, as in a bubble sort.

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What this needs

To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Symmetric_group, Generators and relations
  2. Web https://en.wikipedia.org/wiki/Cyclic_permutation, Properties

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