Finite groups · Family
Simple alternating groups
Aₙ is nonabelian simple for n ≥ 5.
Classification coverageFinite simple classification
Parameter domain
Aₙ is nonabelian simple for n ≥ 5.
| Parameter | Allowed values |
|---|---|
| n | integer n ≥ 5 |
Reference properties
Exceptions and identifications
n = 5: A₅ ≅ PSL(2,4) ≅ PSL(2,5). [sources]
n = 6: A₆ ≅ PSL(2,9) ≅ PSp(4,2)′. [sources]
n = 8: A₈ ≅ PSL(4,2). [sources]
Names and aliases
Alt(n)
Related objects and maps
- A₅ reference
- A₆ reference
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
The subgroup of even permutations in the symmetric group Sₙ.
- Finite simple groups: Lecture 1 — Introduction and alternating groupsCFSG; even permutations; simplicity of AₙOpen source
family:finite-alternating#definitionParameter domain
n: integer n ≥ 5
- Finite simple groups: Lecture 1 — Introduction and alternating groupsCFSG; even permutations; simplicity of AₙOpen source
family:finite-alternating#parametersScope and restrictions
Aₙ is nonabelian simple for n ≥ 5.
- Finite simple groups: Lecture 1 — Introduction and alternating groupsCFSG; even permutations; simplicity of AₙOpen source
family:finite-alternating#restrictionOrder
n! / 2
family:finite-alternating#order-formulaExceptions and identifications
n = 5: A₅ ≅ PSL(2,4) ≅ PSL(2,5).
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-alternating#exceptions-1Exceptions and identifications
n = 6: A₆ ≅ PSL(2,9) ≅ PSp(4,2)′.
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-alternating#exceptions-2Exceptions and identifications
n = 8: A₈ ≅ PSL(4,2).
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-alternating#exceptions-3A₅ reference
group:a-5
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-alternating#relation-1A₆ reference
group:a-6
- Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source
family:finite-alternating#relation-2