S5
E904571
S5 is the symmetric group on five elements, a fundamental non-abelian finite group that plays a key role in permutation group theory and Galois theory.
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
abstract algebraic structure
ⓘ
finite group ⓘ non-abelian group ⓘ permutation group ⓘ symmetric group ⓘ |
| actsOn | set of five elements ⓘ |
| appearsInGaloisTheoryAs | Galois group of many quintic extensions of Q ⓘ |
| containsSubgroup |
A5
ⓘ
C2 ⓘ C5 ⓘ D5 ⓘ S4 ⓘ V4 ⓘ |
| hasAutomorphismGroupIsomorphicTo | S5 ⓘ |
| hasCayleyGraphUsedIn | combinatorics and group theory ⓘ |
| hasCenter | trivial group ⓘ |
| hasConjugacyClass |
2-cycles
GENERATED
ⓘ
3-cycles GENERATED ⓘ 4-cycles GENERATED ⓘ 5-cycles GENERATED ⓘ identity permutation GENERATED ⓘ products of a 3-cycle and a disjoint 2-cycle GENERATED ⓘ products of two disjoint 2-cycles GENERATED ⓘ |
| hasExponent | 60 ⓘ |
| hasIndex | 2 in S5 over A5 ⓘ |
| hasNormalSubgroup | A5 NERFINISHED ⓘ |
| hasNumberOfConjugacyClasses | 7 ⓘ |
| hasOrder | 120 ⓘ |
| hasOrderOfSylow2Subgroup | 8 ⓘ |
| hasOrderOfSylow3Subgroup | 3 ⓘ |
| hasOrderOfSylow5Subgroup | 5 ⓘ |
| hasOuterAutomorphismGroup | trivial group ⓘ |
| hasQuotientIsomorphicTo | C2 ⓘ |
| hasSignHomomorphismTo | C2 ⓘ |
| hasTrivialCenter | true ⓘ |
| isCompleteGroup | true ⓘ |
| isDenotedBy |
Sym(5)
ⓘ
Σ5 ⓘ |
| isDoublyTransitive | true ⓘ |
| isGaloisGroupOf | generic irreducible quintic polynomial over Q ⓘ |
| isGeneratedBy |
(1 2) and (1 2 3 4 5)
ⓘ
a 2-cycle and a 5-cycle ⓘ |
| isGeneratedByTranspositions | true ⓘ |
| isGroupOfPermutationsOf | five-element set ⓘ |
| isIsomorphicTo | group of all bijections on a 5-element set ⓘ |
| isNonAbelian | true ⓘ |
| isNonAbelianFor | n ≥ 3 ⓘ |
| isPrimitivePermutationGroup | true ⓘ |
| isSimple | true ⓘ |
| isTransitiveGroupOn | 5 points ⓘ |
| kernelOfSignHomomorphism | A5 ⓘ |
| quotientBy | A5 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Clebsch diagonal surface