Math 2.0

example

The alternating group A5 is simple

Example

The group A5A_5 has order 6060. Its conjugacy classes have sizes 11, 1515, 2020, 1212 and 1212. The class of size 11 is {e}\{e\}, and the class of size 2020 contains the 33-cycles. The class of size 1515 contains the products of two disjoint transpositions. The 2424 five-cycles split into two classes of size 1212. A normal subgroup NN of A5A_5 is a union of conjugacy classes, and it contains the class {e}\{e\}. By Lagrange's theorem, ∣N∣|N| divides 6060. No sum of 11 and some of the numbers 15,20,12,1215, 20, 12, 12 divides 6060, except 11 and 6060. Thus N={e}N = \{e\} or N=A5N = A_5, and A5A_5 is simple. It is the smallest simple group that is not abelian.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 25 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 levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionLagrange's theorem, statement, no evidence levelLagrange's theoremstatementSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementTransposition, definition, no evidence levelTranspositiondefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementDecomposition of a permutation into transpositions, statement, no evidence levelDecomposition of a pe…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionSign of a permutation, definition, no evidence levelSign of a permutationdefinitionEven and odd permutations, definition, no evidence levelEven and odd permutat…definitionSign of a cycle, statement, no evidence levelSign of a cyclestatementSign is a homomorphism, statement, no evidence levelSign is a homomorphismstatementAlternating group, definition, no evidence levelAlternating groupdefinitionThe alternating group is generated by three-cycles, statement, no evidence levelThe alternating group…statementSimplicity of the alternating group, statement, no evidence levelSimplicity of the alt…statementThe alternating group A5 is simple, example, no evidence levelThe alternating group…example
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. Conjugate elements definition
  4. Group homomorphism definition
  5. Lagrange's theorem statement
  6. Subgroup definition
  7. Symmetric group definition
  8. Kernel of a homomorphism definition
  9. Conjugate subgroup definition
  10. Disjoint cycles commute statement
  11. Transposition definition
  12. Subgroup criterion statement
  13. Decomposition of a permutation into disjoint cycles statement
  14. Normal subgroup definition
  15. Intersection of subgroups is a subgroup statement
  16. Decomposition of a permutation into transpositions statement
  17. Subgroup generated by a subset definition
  18. Sign of a permutation definition
  19. Even and odd permutations definition
  20. Sign of a cycle statement
  21. Sign is a homomorphism statement
  22. Alternating group definition
  23. The alternating group is generated by three-cycles statement
  24. Simplicity of the alternating group statement
  25. The alternating group A5 is simple example

Sources

  1. Web https://en.wikipedia.org/wiki/Alternating_group, Basic properties

What links here

No node links to this node yet.