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 Graded algebra Bialgebra |
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
Famous quotes containing the word group:
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)