Math 2.0

statement proposition

The alternating group is generated by three-cycles

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).

Explanations

L1 human

An even permutation is a chain of swaps, taken two at a time. Each pair of swaps can be written with 33-cycles. So the 33-cycles are enough to build all even permutations.

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What this needs

To read this node, read these 19 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 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 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. 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

What depends on this

3 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Alternating_group, Generators and relations

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