General algebraic systems Definition
Binary operation
Definition
Let be a set. A binary operation on is a map . For , the notation for the image of the pair is .
Remarks
Addition, subtraction and multiplication are binary operations on the set of integers. Subtraction is not a binary operation on the set of natural numbers, because is not in . The condition that is in for all is often called closure.
Where this leads
109 pages build on this one.
- Extensions of solvable groups are solvable
- Quotients of solvable groups are solvable
- Infinite cyclic group
- Commutativity
- Circle group
- Cyclic groups are abelian
- Inner automorphism group is isomorphic to G modulo its centre
- Subgroups of solvable groups are solvable
- Abelian group
- Classification of finite abelian groups
- Group of units of a monoid
- Homomorphisms preserve identity and inverses
- Inverse element
- Additive group of integers
- The kernel is a normal subgroup
- First isomorphism theorem
- Uniqueness of inverses
- Cayley table
- Burnside's p^a q^b theorem
- Automorphism group of a cyclic group
- Nilpotent groups are solvable
- Units of Z/nZ are the residues coprime to n
- 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
- A homomorphism is injective iff its kernel is trivial
- Associativity
- Solvable group
- Klein four-group
- Second isomorphism theorem
- The group of units of Z/pZ is cyclic
- Abelian groups are solvable
- Monoid
- Torsion subgroup
- Identity element
- Structure theorem for finitely generated abelian groups
- Order of the group of units of Z/nZ is Euler's totient
- Subgroups of the integers
- Order of the alternating group
- Third isomorphism theorem
- Semigroup
- Cancellation laws
- Group of nth roots of unity
- Integers modulo n as a quotient group
- Free abelian group
- Abel–Ruffini theorem
- Solenoid
- Zassenhaus lemma
- 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
- Proof that the kernel is a normal subgroup
- Wiener–Hopf factorization
- Archimedean equivalence classes
- Proof that a homomorphism is injective iff its kernel is trivial
- Newton polygon
- The Klein four-group is abelian but not cyclic
- Fourier monomial on the circle
- Quantale
- 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
- Proof of the first isomorphism theorem
- Milnor K-theory
- Wiener–Hopf method
- Combinatorics on words
- Wiener's lemma
- Algebraic cycle
- Algebraic K-theory of schemes
- Near-ring
- Structure theorem for compactly generated LCA groups
- Left-orderable group
- Schreier refinement theorem
- Solvability of x^n - a
- The natural numbers under addition are not a group
- Ordered semigroup
- Proof of the uniqueness of inverses
- Quintic equation
- Residuated lattice
- 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
- Proof of the cancellation laws
- Orderable group
- Étale cohomology
Further reading
- Web https://en.wikipedia.org/wiki/Binary_operation, lead section
About this page
| Tag | 0580ZJSA3Q |
|---|---|
| URI | m2://0580ZJSA3Q |
| Kind | definition |
| Writing | W1 stated |
| MSC | 08A02 |
| Defines | Binary operation |
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