Abelian group

Basic properties and notation

A set G with operation ⋆ is an abelian group when it satisfies: closure, associativity, an identity element, and existence of inverses for all elements, together with commutativity. Many authors adopt additive notation for abelian groups and write the operation as +, the identity as 0, and inverses as −a. Elementary consequences include:

• Every subgroup and every quotient of an abelian group is abelian.
• The commutator subgroup is trivial if and only if the group is abelian.
• Finite direct products of abelian groups are abelian.

Common examples

Examples that illustrate the range of abelian groups include:

• The integers Z under addition, and similarly the additive groups of rational, real or complex numbers.
• Finite cyclic groups, such as Z/nZ, generated by a single element of finite order.
• Vector spaces over a field, viewed as abelian groups under vector addition.
• Groups of points on an elliptic curve (important in number theory and cryptography), which are abelian under the chord-and-tangent law.

Structure and classification

Finitely generated abelian groups admit a precise classification: each such group is isomorphic to a direct sum of a free part (copies of the integers Z) and a finite torsion part built from cyclic groups. This statement is commonly presented either in the invariant factor form or the primary decomposition form; both describe how the group decomposes into simpler cyclic components. The classification is a foundational result used to reduce problems about abelian groups to problems about Z and finite cyclic groups.

History and context

The name "abelian" honors Niels Henrik Abel, a 19th-century Norwegian mathematician whose work on equations and algebraic structures influenced the study of commutativity in algebra. The concept became a standard object of study in modern algebra and provides the simplest nontrivial setting in which many ideas—such as homomorphisms, exact sequences and module theory—can be developed and tested.

Uses and distinctions

Abelian groups appear throughout mathematics: as homology groups in algebraic topology, class groups in algebraic number theory, additive groups of rings, and as modules over the integers. They contrast with non-abelian groups, where order matters and many familiar simplifications fail. Recognizing when a group is abelian simplifies both computations and theory: many proofs that are hard or false for general groups become elementary in the commutative setting. For introductory material and connections to broader group theory, see the standard references in group theory texts.