Alphabeta MathHow statement and proof provenance work
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The alternating group of even permutations
Definition
For , the alternating group is the kernel of the sign homomorphism,
Thus consists exactly of the even permutations. The subgroup and normality assertions implicit in the word “group” follow from The image of a group homomorphism is a subgroup and its kernel is a normal subgroup.
Depends on
Used by
- [Sₙ,Sₙ]=Aₙ for n≥2, and [Aₙ,Aₙ]=Aₙ for n≥5 Corollary
- Aₙ is normal in Sₙ; for n≥2, 2 |Aₙ|=n!, while Aₙ=Sₙ for n=0,1 Corollary
- For n≥5, the only proper nontrivial normal subgroup of Sₙ is Aₙ Corollary
- A₄ has four complements to its normal Klein four subgroup Example
- A₄ is not nilpotent Example
- For n≥2, Sₙ≅ Aₙ⋊ C₂ using any transposition complement Example
- The six elements of S₃: one-line form, cycle structure, inversions, and sign Example
- The Sylow subgroups of A₅ Example
- The three transposition subgroups of S₃ are conjugate complements to A₃ Example
- V₄={1,(12)(34),(13)(24),(14)(23)} is a proper nontrivial normal subgroup of A₄ Example
- False statement: every divisor of the order of a finite group occurs as a subgroup order False statement
- FALSE: Aₙ is simple for every n≥4 False statement
- FALSE: two even permutations of the same cycle type are always conjugate in Aₙ False statement
- The symmetric groups Sₙ are solvable for n≤ 4 Lemma
- The transitive subgroups of S₄ and their action on the three pairings Lemma
- A monic irreducible separable cubic in characteristic not two has Galois group A₃ or S₃ according to its discriminant Theorem
- Aₙ is generated by 3-cycles for every n≥3 Theorem
- Aₙ is simple for every n≥5 Theorem
- For a monic separable polynomial in characteristic not two, the Galois group lies in Aₙ exactly when the discriminant is a square Theorem
- For n≥2, an Sₙ-class of an even permutation splits in Aₙ exactly when all cycle lengths, including 1-cycles, are odd and distinct Theorem
- The cycle index of Aₙ is the parity-filtered symmetric-group sum Theorem
Dependency tree · two levels
12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- T. W. Judson, Abstract Algebra: Theory and Applications, §5.1 (standard reference, not scraped)