Math 2.0

example

The alternating group is the kernel of the sign

Example

Let n≥2n \ge 2. The sign is a homomorphism sgn⁡:Sn→{1,−1}\operatorname{sgn} : S_n \to \{1, -1\}, where {1,−1}\{1, -1\} is a group under multiplication. A permutation has sign 11 when it is a product of an even number of transpositions. Thus the kernel of sgn⁡\operatorname{sgn} is the set of even permutations. This is the alternating group AnA_n. Thus AnA_n is a normal subgroup of SnS_n.

Remarks

The sign is onto, so AnA_n has index 22 in SnS_n.

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To read this node, read these 11 nodes in this order. The foundations come first, and the list ends with this node.

Alternating group, concept, no evidence levelAlternating groupconceptCycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptSymmetric group, concept, no evidence levelSymmetric groupconceptSymmetric group, definition, no evidence levelSymmetric groupdefinitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementTransposition, definition, no evidence levelTranspositiondefinitionDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementDecomposition of a permutation into transpositions, statement, no evidence levelDecomposition of a pe…statementSign of a permutation, definition, no evidence levelSign of a permutationdefinitionThe alternating group is the kernel of the sign, 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. Alternating group concept
  2. Cycle notation concept
  3. Group concept
  4. Symmetric group concept
  5. Symmetric group definition
  6. Disjoint cycles commute statement
  7. Transposition definition
  8. Decomposition of a permutation into disjoint cycles statement
  9. Decomposition of a permutation into transpositions statement
  10. Sign of a permutation definition
  11. The alternating group is the kernel of the sign example

Sources

  1. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, Properties

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