Group theory and generalizations Foundations Concept
Abelian group
Summary
An abelian group is a group whose operation is commutative. The integers with addition are the basic example.
Where this leads
76 pages build on this one.
- Grothendieck category
- Semifield
- Group cohomology
- Near-field
- Sumset
- Eilenberg–Steenrod axioms
- Biproduct
- Sheaf of modules
- Singular cohomology
- Kummer theory
- Summation in topological groups
- Artin reciprocity law
- Witt group
- Sheaf
- Additive energy
- Elementary abelian group
- Weil divisor
- Category of abelian groups
- Class field theory
- Subcategory
- Kronecker–Weber theorem
- Additive function
- Rank of an abelian group
- Generator
- Mordell's theorem
- Bredon cohomology
- Cohomology operation
- Sheaf cohomology
- Ideal class group
- Borel–Moore homology
- Graded ring
- Characteristic class
- Ulm's theorem
- Plünnecke–Ruzsa inequality
- Local coefficient system
- Dehn invariant
- Whitehead problem
- Character of an abelian group
- Section and retraction
- Group law on an elliptic curve
- Hilbert class field
- Additive functor
- Gowers norm
- Grothendieck group
- Serre's twisting sheaf
- Picard group
- Cokernel
- Higher homotopy groups are abelian
- Sum-free set
- Congruent number problem
- Mordell–Weil theorem
- Preadditive category
- Tate module
- Additive combinatorics
- Differential graded algebra
- Valuation on convex bodies
- Ruzsa triangle inequality
- Divisible group
- Alexander–Spanier cohomology
- Approximately additive function
- Čech cohomology
- Sheaf cohomology in topology
- Toyoda–Bruck theorem
- Coproduct
- Balog–Szemerédi–Gowers theorem
- Sidon set
- Eilenberg–MacLane space
- Cobordism ring
- Quadratic functional equation
- Jordan–Schur theorem
- Full subcategory
- Hodge structure
- Homology with coefficients
- Generalized cohomology theory
- Locally compact abelian group
- Birch and Swinnerton-Dyer conjecture
Further reading
- Web https://en.wikipedia.org/wiki/Abelian_group, Definition
About this page
| Tag | 082GHNPJ53 |
|---|---|
| URI | m2://082GHNPJ53 |
| Aliases | commutative-group |
| Kind | concept |
| Writing | W2 usable |
| MSC | 20Kxx |
| Specializes | Group |
| Same as |
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What this page needs for W3
- section Motivation: the question that the notion answers (0 of 1)
- section History: who introduced the notion and when (0 of 1)
- explanations for the audience (0 of 3)
- a person confirmed each agent-written section: rule P04 (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Instances
- Circle group
- Additive group of integers
- Group of units of Z/nZ
- Group of nth roots of unity
- The Klein four-group is abelian but not cyclic
- Integers modulo n form a cyclic group
Results about this
Definitions
Special cases
Used by
- Grothendieck category
- Semifield
- Group cohomology
- Near-field
- Sumset
- Eilenberg–Steenrod axioms
- Biproduct
- Sheaf of modules
- Singular cohomology
- Kummer theory
- Summation in topological groups
- Artin reciprocity law
- Witt group
- Sheaf
- Additive energy
- Elementary abelian group
- Weil divisor
- Category of abelian groups
- Class field theory
- Subcategory
- Kronecker–Weber theorem
- Additive function
- Rank of an abelian group
- Generator
- Mordell's theorem
- Bredon cohomology
- Cohomology operation
- Sheaf cohomology
- Ideal class group
- Borel–Moore homology
- Graded ring
- Characteristic class
- Ulm's theorem
- Plünnecke–Ruzsa inequality
- Local coefficient system
- Dehn invariant
- Whitehead problem
- Character of an abelian group
- Section and retraction
- Group law on an elliptic curve
- Hilbert class field
- Additive functor
- Gowers norm
- Grothendieck group
- Picard group
- Cokernel
- Higher homotopy groups are abelian
- Sum-free set
- Mordell–Weil theorem
- Preadditive category
- Tate module
- Additive combinatorics
- Valuation on convex bodies
- Ruzsa triangle inequality
- Divisible group
- Alexander–Spanier cohomology
- Approximately additive function
- Čech cohomology
- Sheaf cohomology in topology
- Toyoda–Bruck theorem
- Coproduct
- Balog–Szemerédi–Gowers theorem
- Sidon set
- Eilenberg–MacLane space
- Quadratic functional equation
- Jordan–Schur theorem
- Full subcategory
- Hodge structure
- Homology with coefficients
- Generalized cohomology theory
- Locally compact abelian group