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
Commutative ring
Integral domain
Lattice-like structures Semilattice
Map of lattices
Module-like structures Group with operators
Vector space
Algebra-like structures Algebra
Associative algebra
Non-associative algebra
Graded algebra

Read more about Abelian Group:  Definition, Examples, Historical Remarks, Properties, Finite Abelian Groups, Infinite Abelian Groups, Relation To Other Mathematical Topics, A Note On The Typography

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