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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Abelian groups canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T511394 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Abelian groups Context triple: [Niels Henrik Abel, knownFor, Abelian groups]
-
A.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
B.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
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C.
Conway groups
Conway groups are a set of three closely related sporadic simple groups discovered by John H. Conway in the study of symmetries of the Leech lattice in group theory.
-
D.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
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E.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Abelian groups Target entity description: Abelian groups are algebraic structures in which the group operation is commutative, meaning the order of combining elements does not affect the result.
-
A.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
B.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
C.
Conway groups
Conway groups are a set of three closely related sporadic simple groups discovered by John H. Conway in the study of symmetries of the Leech lattice in group theory.
-
D.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
-
E.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Abelian groups Description of subject: Abelian groups are algebraic structures in which the group operation is commutative, meaning the order of combining elements does not affect the result.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.