# Abelian Group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel.

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research.

Algebraic structures
Group-like structures Semigroup and Monoid
Quasigroup and Loop
Abelian group
Ring-like structures Semiring
Near-ring
Ring
Commutative ring
Integral domain
Field
Lattice-like structures Semilattice
Lattice
Map of lattices
Module-like structures Group with operators
Module
Vector space
Algebra-like structures Algebra
Associative algebra
Non-associative algebra
Bialgebra

### Other articles related to "abelian group, group, abelian groups, abelian":

Tensor Product Of Modules - Multilinear Mappings
... a right R-module MR, a left R-module RN, and an abelian group Z, a bilinear map or balanced product from M × N to Z is a function φ M × N → Z ... This is necessary because Z is only assumed to be an abelian group, so r·φ(m,n) would not make sense ... This turns the set Bilin(M,NZ) into an abelian group ...
Stone–von Neumann Theorem - Uniqueness of Representation - Representation Theory Formulation
... representations of the Heisenberg group ... This is discussed in more detail in the Heisenberg group section, below ... technical assumptions, every representation of the Heisenberg group is equivalent to the position operators and momentum operators on Rn ...
Surgery Exact Sequence - Versions Revisited
... This is achieved by putting an alternative abelian group structure on the normal invariants as described here ... with the algebraic surgery exact sequence of Ranicki which is an exact sequence of abelian groups by definition ... This gives the structure set the structure of an abelian group ...
Sum-free Set
... and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A⊕A is disjoint from A ... How many sum-free sets does an abelian group G contain? What is the size of the largest sum-free set that an abelian group G contains? A sum-free set is said to be maximal if it is not a proper subset of another ...
Abelian Group - A Note On The Typography
... adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, indicating ...

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