Finite groups · Family
Finite symmetric groups
All permutations of n symbols; order n!.
Reference coverageFinite families
Definition
The group of bijections of an n-element set, with composition as the operation.
Definition sourcesParameter domain
All permutations of n symbols; order n!.
| Parameter | Allowed values |
|---|---|
| n | positive integer |
Reference properties
Conventions and use
The sign homomorphism has kernel Aₙ; for n ≥ 5 that kernel is nonabelian simple. [sources]
Names and aliases
Sym(n)
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 group of bijections of an n-element set, with composition as the operation.
- Finite simple groups: Lecture 1 — Introduction and alternating groupsPermutations; even permutationsOpen source
family:finite-symmetric#definitionParameter domain
n: positive integer
- Finite simple groups: Lecture 1 — Introduction and alternating groupsPermutations; even permutationsOpen source
family:finite-symmetric#parametersScope and restrictions
All permutations of n symbols; order n!.
- Finite simple groups: Lecture 1 — Introduction and alternating groupsPermutations; even permutationsOpen source
family:finite-symmetric#restrictionConventions and use
The sign homomorphism has kernel Aₙ; for n ≥ 5 that kernel is nonabelian simple.
- Finite simple groups: Lecture 1 — Introduction and alternating groupsEven permutations; simplicity of alternating groupsOpen source
family:finite-symmetric#conventions-1