Math 2.0

Group theory and generalizations Example

The alternating group is the kernel of the sign

Builds on Sign of a permutationAlternating groupSymmetric group

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.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
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 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. 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

Further reading

  1. Web https://en.wikipedia.org/wiki/Parity_of_a_permutation, Properties
About this page
TagTJ6EBSS3S7
URIm2://TJ6EBSS3S7
Kindexample
WritingW0 stub
MSC20B30
Instance ofKernel of a homomorphism
IllustratesThe kernel is a normal subgroup
UsesSign of a permutation, Alternating group, Symmetric group

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