Abelian groups are named for a Norwegian teen prodigy
When order does not matter under a group operation, mathematicians call the group abelian. Integers under addition are the everyday model. Camille Jordan borrowed the name from Niels Henrik Abel, who linked commutativity of a polynomial's group to roots expressible by radicals.
An abelian—or commutative—group is a group whose binary operation treats a then b the same as b then a. With addition, the integers and the real numbers are classic examples; the idea generalizes those familiar settings. Fields, rings, vector spaces, and algebras all rest on abelian additive structure. The theory is usually milder than for non-abelian groups, and finite abelian groups are completely classified.
Notation splits into additive and multiplicative habits. Multiplicative writing is the default for groups in general; additive writing is common for modules and rings, and sometimes used to signal that a group under discussion is abelian. For a finite group, a Cayley table acts like a multiplication chart: the group is abelian exactly when that table is symmetric across the main diagonal. Every ring is an abelian group under addition; in a commutative ring the units form an abelian group under multiplication. Subgroups of abelian groups are automatically normal, and subgroups, quotients, and direct sums of abelian groups stay abelian. The finite simple abelian groups are precisely the cyclic groups of prime order.
Every abelian group is a module over the integers, and many theorems about abelian groups specialize results about modules over a principal ideal domain. Finitely generated abelian groups split as a direct sum of a torsion piece and a free abelian piece. Rank measures a maximal linearly independent set over the integers: finite and torsion groups have rank zero; the integers and the rationals have rank one; the multiplicative group of nonzero rationals has infinite rank, freely generated by the primes via unique factorization. Matrices usually fail to form an abelian multiplicative group, though special families can. Jordan's naming honors Abel's insight that commutativity of a polynomial's Galois group lets roots be extracted by radicals.
Source: Abelian group
