Group theory and generalizations Definition
Abelian group
Builds on GroupCommutativity
Definition
An abelian group is a group whose operation is commutative. Thus for all .
Remarks
The integers, the rationals and the real numbers with addition are abelian groups. The operation of an abelian group is often written , with identity and inverse . The group of invertible real matrices under multiplication is not abelian.
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
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- Binary operation definition
- Group concept
- Commutativity definition
- Abelian group definition
Where this leads
73 pages build on this one.
- Extensions of solvable groups are solvable
- Quotients of solvable groups are solvable
- Infinite cyclic group
- Circle group
- Cyclic groups are abelian
- Subgroups of solvable groups are solvable
- Classification of finite abelian groups
- Additive group of integers
- Burnside's p^a q^b theorem
- Nilpotent groups are solvable
- The symmetric group on five or more letters is not solvable
- Subgroups of abelian groups are normal
- A finite abelian group is cyclic iff its exponent equals its order
- Solvable group
- Klein four-group
- Abelian groups are solvable
- Torsion subgroup
- Structure theorem for finitely generated abelian groups
- Subgroups of the integers
- Group of nth roots of unity
- Integers modulo n as a quotient group
- Free abelian group
- Abel–Ruffini theorem
- Solenoid
- Matrix Wiener–Hopf factorization
- Matrix Riemann–Hilbert problem
- Norm residue isomorphism theorem
- Ring spectrum
- Table of Lie groups
- Fitting subgroup
- Proof that cyclic groups are abelian
- Loop group
- Rng
- Wiener–Hopf factorization
- Archimedean equivalence classes
- Newton polygon
- The Klein four-group is abelian but not cyclic
- Fourier monomial on the circle
- Galois cohomology
- Pontryagin dual
- Brauer group
- Exact couple
- Hahn embedding theorem
- Character of an abelian group
- Lattice-ordered group
- Representability of cohomology by Eilenberg–MacLane spaces
- Milnor K-theory
- Wiener–Hopf method
- Wiener's lemma
- Algebraic cycle
- Algebraic K-theory of schemes
- Structure theorem for compactly generated LCA groups
- Left-orderable group
- Solvability of x^n - a
- Quintic equation
- Shafarevich's theorem on solvable Galois groups
- Abelian extension
- Valuation
- Hahn series
- Bockstein homomorphism
- Galois' theorem on solvability
- Torsion-free group
- Nonassociative ring
- Eilenberg–MacLane spectrum
- Cohomological dimension of a field
- Torus as a group
- List of small groups
- Divisor class group
- Trigonometric moment problem
- Maximal torus
- Solvable by radicals
- Orderable group
- Étale cohomology
Further reading
- Web https://en.wikipedia.org/wiki/Abelian_group, Definition
About this page
| Tag | 071DDFVG4J |
|---|---|
| URI | m2://071DDFVG4J |
| Kind | definition |
| Writing | W2 usable |
| MSC | 20Kxx |
| Defines | Abelian group |
| Uses | Group, Commutativity |
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What links here
Used by
- Circle group
- Cyclic groups are abelian
- Classification of finite abelian groups
- Additive group of integers
- Subgroups of abelian groups are normal
- A finite abelian group is cyclic iff its exponent equals its order
- Solvable group
- Klein four-group
- Abelian groups are solvable
- Torsion subgroup
- Structure theorem for finitely generated abelian groups
- Group of nth roots of unity
- Free abelian group
- Proof that cyclic groups are abelian
- Rng
- Archimedean equivalence classes
- Galois cohomology
- Brauer group
- Exact couple
- Hahn embedding theorem
- Representability of cohomology by Eilenberg–MacLane spaces
- Milnor K-theory
- Abelian extension
- Valuation
- Hahn series
- Bockstein homomorphism
- Nonassociative ring
- Eilenberg–MacLane spectrum
- Orderable group