Abelian groups are named after the Norwegian mathematician Niels Henrik Abel.
Camille Jordan named abelian groups after the Norwegian mathematician Niels Henrik Abel, who had found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals.
) can often be generalized to theorems about modules over an arbitrary principal ideal domain.
It is defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group.
On the other hand, the multiplicative group of the nonzero rationals has an infinite rank, as it is a free abelian group with the set of the prime numbers as a basis (this results from the fundamental theorem of arithmetic).
It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants.
The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.
can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups.
The classification was proven by Leopold Kronecker in 1870, though it was not stated in modern group-theoretic terms until later, and was preceded by a similar classification of quadratic forms by Carl Friedrich Gauss in 1801; see history for details.
is isomorphic to a direct sum of the form in either of the following canonical ways: For example,
One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group
of subgroups of coprime order, then Given this, the fundamental theorem shows that to compute the automorphism group of
to be of the form so elements of this subgroup can be viewed as comprising a vector space of dimension
, and One can check that this yields the orders in the previous examples as special cases (see Hillar & Rhea).
such that This homomorphism is surjective, and its kernel is finitely generated (since integers form a Noetherian ring).
Changing the generating set of the kernel of M is equivalent with multiplying M on the right by a unimodular matrix.
The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums.
, and the prime powers giving the orders of finite cyclic summands are uniquely determined.
By contrast, classification of general infinitely generated abelian groups is far from complete.
, constitute one important class of infinite abelian groups that can be completely characterized.
An abelian group is called periodic or torsion, if every element has finite order.
In a different direction, Helmut Ulm found an extension of the second Prüfer theorem to countable abelian
-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.
[18]: 317 An abelian group is called torsion-free if every non-zero element has infinite order.
The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups.
Important technical tools used in classification of infinite abelian groups are pure and basic subgroups.
See the books by Irving Kaplansky, László Fuchs, Phillip Griffith, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in Lecture Notes in Mathematics for more recent findings.
Wanda Szmielew (1955) proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable.
There are still many areas of current research: Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics.
In the 1970s, Saharon Shelah proved that the Whitehead problem is: Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, the lack of capitalization being a tacit acknowledgment not only of the degree to which Abel's name has been institutionalized but also of how ubiquitous in modern mathematics are the concepts introduced by him.