Abelian groups
E63711
Abelian groups are algebraic structures in which the group operation is commutative, meaning the order of combining elements does not affect the result.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Abelian group | 0 |
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
group ⓘ |
| alsoKnownAs | commutative group ⓘ |
| closedUnder |
finite sums of elements
ⓘ
taking inverses ⓘ |
| example |
complex numbers under addition
ⓘ
cyclic group of order n ⓘ finite direct product of cyclic groups ⓘ integers under addition ⓘ rationals under addition ⓘ reals under addition ⓘ torsion subgroup of a group ⓘ vector space under addition ⓘ |
| generalizes |
additive group of a ring
ⓘ
additive group of a vector space ⓘ cyclic group ⓘ |
| hasAxiom |
associativity of group operation
ⓘ
closure under group operation ⓘ commutativity of group operation ⓘ existence of identity element ⓘ existence of inverses ⓘ |
| hasCategory | category of Abelian groups ⓘ |
| hasCategoryProperty | category of Abelian groups is an Abelian category ⓘ |
| hasConstruction |
direct product of Abelian groups
ⓘ
direct sum of Abelian groups ⓘ quotient group ⓘ |
| hasIdentityElement |
0 in additive notation
ⓘ
e in multiplicative notation ⓘ |
| hasInvariant |
primary decomposition for finite Abelian groups
ⓘ
rank of an Abelian group ⓘ torsion subgroup ⓘ |
| hasInverseNotation |
a⁻¹ in multiplicative notation
ⓘ
−a in additive notation ⓘ |
| hasMorphism | group homomorphism ⓘ |
| hasOperation | binary operation usually denoted by + or · ⓘ |
| hasProperty |
associative operation
ⓘ
commutative operation ⓘ identity element ⓘ inverse elements ⓘ |
| hasStructureTheorem | finitely generated Abelian groups decompose into direct sum of cyclic groups ⓘ |
| hasSubstructure | subgroup ⓘ |
| isSpecialCaseOf | group ⓘ |
| namedAfter | Niels Henrik Abel ⓘ |
| satisfiesEquation | a + b = b + a for all elements a, b ⓘ |
| usedIn |
algebraic number theory
ⓘ
algebraic topology ⓘ category theory ⓘ homological algebra ⓘ module theory ⓘ representation theory ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.