Math 2.0

Group theory and generalizations Definition

Alternating group

Builds on Even and odd permutationsSign is a homomorphismKernel of a homomorphism

Definition

Let nn be a positive integer. The alternating group AnA_n is the set of the even permutations in SnS_n, with composition as the operation.

Remarks

For n≥2n \geq 2, AnA_n is the kernel of the sign homomorphism, so it is a normal subgroup of SnS_n. For n≥2n \geq 2, AnA_n is also the commutator subgroup of SnS_n. The group A3A_3 is cyclic of order 33, and A4A_4 has order 1212. The group AnA_n is abelian if and only if n≤3n \leq 3.

Reading path

To understand this page from the ground up, read these 13 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 homomorphismdefinitionSymmetric 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…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 permutationdefinitionEven and odd permutations, definition, no evidence levelEven and odd permutat…definitionSign is a homomorphism, statement, no evidence levelSign is a homomorphismstatementAlternating group, definition, no evidence levelAlternating groupdefinition
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. Symmetric group definition
  5. Kernel of a homomorphism 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. Even and odd permutations definition
  12. Sign is a homomorphism statement
  13. Alternating group definition

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Alternating_group, lead section
  2. Web https://en.wikipedia.org/wiki/Alternating_group, Basic properties
About this page
Tag0T13HM377F
URIm2://0T13HM377F
Kinddefinition
WritingW1 stated
MSC20B30
DefinesAlternating group
UsesEven and odd permutations, Sign is a homomorphism, Kernel of a homomorphism

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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

What links here

Used by