General algebraic systems Definition
Commutativity
Builds on Binary operation
Definition
Let be a set and let be a binary operation on . The operation is commutative when for all .
Remarks
Addition and multiplication of real numbers are commutative. Subtraction is not commutative: . Multiplication of real matrices is associative but not commutative.
Reading path
To understand this page from the ground up, read these 2 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Binary operation definition
- Commutativity definition
Where this leads
74 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
- Abelian group
- 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
About this page
| Tag | 04E4V9G5FR |
|---|---|
| URI | m2://04E4V9G5FR |
| Kind | definition |
| Writing | W1 stated |
| MSC | 08A02 |
| Defines | Commutativity |
| Uses | Binary operation |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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