Math 2.0

Group theory and generalizations Proposition

The alternating group is generated by three-cycles

Builds on Alternating groupSubgroup generated by a subsetSign of a cycle

Statement

Let n≥3n \geq 3. The alternating group AnA_n is generated by the 33-cycles in SnS_n.

Remarks

Each 33-cycle is even, so the 33-cycles lie in AnA_n. An even permutation is a product of an even number of transpositions, and each product of two transpositions is a product of 33-cycles. For distinct a,b,c,da, b, c, d, the formulas are (a b)(a c)=(a c b)(a\ b)(a\ c) = (a\ c\ b) and (a b)(c d)=(a c b)(a c d)(a\ b)(c\ d) = (a\ c\ b)(a\ c\ d).

Reading path

To understand this page from the ground up, read these 19 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 homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementSubgroup criterion, statement, no evidence levelSubgroup criterionstatementTransposition, definition, no evidence levelTranspositiondefinitionDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionDecomposition 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…definitionSign is a homomorphism, statement, no evidence levelSign is a homomorphismstatementSign of a cycle, statement, no evidence levelSign of a cyclestatementAlternating group, definition, no evidence levelAlternating groupdefinitionThe alternating group is generated by three-cycles, statement, no evidence levelThe alternating group…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. Group homomorphism definition
  4. Subgroup definition
  5. Symmetric group definition
  6. Kernel of a homomorphism definition
  7. Disjoint cycles commute statement
  8. Subgroup criterion statement
  9. Transposition definition
  10. Decomposition of a permutation into disjoint cycles statement
  11. Intersection of subgroups is a subgroup statement
  12. Subgroup generated by a subset definition
  13. Decomposition of a permutation into transpositions statement
  14. Sign of a permutation definition
  15. Even and odd permutations definition
  16. Sign is a homomorphism statement
  17. Sign of a cycle statement
  18. Alternating group definition
  19. The alternating group is generated by three-cycles statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Alternating_group, Generators and relations
About this page
Tag0KFZ0QX83G
URIm2://0KFZ0QX83G
Kindstatement, proposition
WritingW0 stub
MSC20B30
StatusThis copy of the site has no build data. Run m2 status 0KFZ0QX83G.
UsesAlternating group, Subgroup generated by a subset, Sign of a cycle

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