abeliangroup

An abelian group is a mathematical structure consisting of a set G equipped with a binary operation, often written as addition (+) or multiplication (·), that satisfies four core properties: associativity (a + (b + c) = (a + b) + c), a unit element e (there exists e such that e + a = a for all a in G), inverses (for every a there is b with a + b = e), and commutativity (a + b = b + a for all a, b in G). The commutativity condition distinguishes abelian groups from general groups.

The term is named after the Norwegian mathematician Niels Henrik Abel. In practice, abelian groups are typically denoted additively, with the identity element written as 0 and inverses written as -a, although multiplicative notation is also common.

Common examples include the integers under addition (Z, +), the rational, real, and complex numbers under addition, and finite cyclic groups such as Z_n (integers modulo n) under addition. The multiplicative group of nonzero real or complex numbers is abelian as well (R^× and C^×). In contrast, the symmetric group S_n for n ≥ 3 is non-abelian.

Key concepts associated with abelian groups include subgroups (which are automatically normal in abelian groups), quotient groups, and homomorphisms that preserve the group operation. Elements of finite order (torsion elements) have a finite number of multiples that yield the identity; many abelian groups are described in terms of direct sums or products.

A fundamental result for finitely generated abelian groups states that any such group is isomorphic to a direct sum of a free part Z^r and a finite torsion part, providing a precise classification up to isomorphism. This framework underpins much of algebra and number theory.