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 the order in which they are written (the axiom of commutativity).

Description

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, from which many other basic concepts, such as modules and vector spaces are developed.

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.

See also

• Abelianization
• Class field theory
• Commutator subgroup
• Dihedral group of order 6, the smallest non-Abelian group
• Elementary abelian group
• Pontryagin duality
• Pure injective module
• Pure projective module

External links

• [ Abelian group] @ Wikipedia.org