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In mathematics, 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. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.^{ [1] }

- Definition
- Facts
- Notation
- Multiplication table
- Examples
- Historical remarks
- Properties
- Finite abelian groups
- Classification
- Automorphisms
- Finitely generated abelian groups
- Infinite abelian groups
- Torsion groups
- Torsion-free and mixed groups
- Invariants and classification
- Additive groups of rings
- Relation to other mathematical topics
- A note on the typography
- See also
- Notes
- References
- External links

The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.

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Totality ^{ α } | Associativity | Identity | Invertibility | Commutativity | |

Semigroupoid | Unneeded | Required | Unneeded | Unneeded | Unneeded |

Small Category | Unneeded | Required | Required | Unneeded | Unneeded |

Groupoid | Unneeded | Required | Required | Required | Unneeded |

Magma | Required | Unneeded | Unneeded | Unneeded | Unneeded |

Quasigroup | Required | Unneeded | Unneeded | Required | Unneeded |

Unital Magma | Required | Unneeded | Required | Unneeded | Unneeded |

Loop | Required | Unneeded | Required | Required | Unneeded |

Semigroup | Required | Required | Unneeded | Unneeded | Unneeded |

Inverse Semigroup | Required | Required | Unneeded | Required | Unneeded |

Monoid | Required | Required | Required | Unneeded | Unneeded |

Commutative monoid | Required | Required | Required | Unneeded | Required |

Group | Required | Required | Required | Required | Unneeded |

Abelian group | Required | Required | Required | Required | Required |

^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently. |

An abelian group is a set, , together with an operation that combines any two elements and of to form another element of denoted . The symbol is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, , must satisfy five requirements known as the *abelian group axioms*:

- Closure
- For all , in , the result of the operation is also in .
- Associativity
- For all , , and in , the equation holds.
- Identity element
- There exists an element in , such that for all elements in , the equation holds.
- Inverse element
- For each in there exists an element in such that , where is the identity element.
- Commutativity
- For all , in , .

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".^{ [2] }^{:11}

There are two main notational conventions for abelian groups – additive and multiplicative.

Convention | Operation | Identity | Powers | Inverse |
---|---|---|---|---|

Addition | 0 | |||

Multiplication | or | 1 |

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules and rings. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being near-rings and partially ordered groups, where an operation is written additively even when non-abelian.^{ [3] }^{:28–29}

To verify that a finite group is abelian, a table (matrix) – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is under the operation , the -th entry of this table contains the product .

The group is abelian if and only if this table is symmetric about the main diagonal. This is true since the group is abelian iff for all , which is iff the entry of the table equals the entry for all , i.e. the table is symmetric about the main diagonal.

- For the integers and the operation addition , denoted , the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer has an additive inverse, , and the addition operation is commutative since for any two integers and .
- Every cyclic group is abelian, because if , are in , then . Thus the integers, , form an abelian group under addition, as do the integers modulo , .
- Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
- Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order.
^{ [4] } - The concepts of abelian group and -module agree. More specifically, every -module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers in a unique way.

In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of rotation matrices.

Camille Jordan named abelian groups after Norwegian mathematician Niels Henrik Abel, because Abel found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals.^{ [5] }^{:144–145}

If is a natural number and is an element of an abelian group written additively, then can be defined as ( summands) and . In this way, becomes a module over the ring of integers. In fact, the modules over can be identified with the abelian groups.

Theorems about abelian groups (i.e. modules over the principal ideal domain ) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form for prime, and the latter is a direct sum of finitely many copies of .

If are two group homomorphisms between abelian groups, then their sum , defined by , is again a homomorphism. (This is not true if is a non-abelian group.) The set of all group homomorphisms from to is therefore an abelian group in its own right.

Somewhat akin to the dimension of vector spaces, every abelian group has a * rank *. It is defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group.^{ [6] }^{:49–50} Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero is a torsion group. The integers and the rational numbers have rank one, as well as every nonzero additive subgroup of the rationals. 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).

The center of a group is the set of elements that commute with every element of . A group is abelian if and only if it is equal to its center . The center of a group is always a characteristic abelian subgroup of . If the quotient group of a group by its center is cyclic then is abelian.^{ [7] }

Cyclic groups of integers modulo , , were among the first examples of groups. 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 automorphism group of a finite abelian group can be described directly in terms of these 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.

Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is also abelian.^{ [8] } In fact, for every prime number there are (up to isomorphism) exactly two groups of order , namely and .

The **fundamental theorem of finite abelian groups** states that every finite abelian group 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**. Moreover, automorphism groups of cyclic groups are examples of abelian groups.^{ [9] } This is generalized by the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when *G* has zero rank; this in turn admits numerous further generalizations.

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.

The cyclic group of order is isomorphic to the direct sum of and if and only if and are coprime. It follows that any finite abelian group is isomorphic to a direct sum of the form

in either of the following canonical ways:

- the numbers are powers of (not necessarily distinct) primes,
- or divides , which divides , and so on up to .

For example, can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: . The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic.

For another example, every abelian group of order 8 is isomorphic to either (the integers 0 to 7 under addition modulo 8), (the odd integers 1 to 15 under multiplication modulo 16), or .

See also list of small groups for finite abelian groups of order 30 or less.

One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group . To do this, one uses the fact that if splits as a direct sum of subgroups of coprime order, then

Given this, the fundamental theorem shows that to compute the automorphism group of it suffices to compute the automorphism groups of the Sylow -subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of ). Fix a prime and suppose the exponents of the cyclic factors of the Sylow -subgroup are arranged in increasing order:

for some . One needs to find the automorphisms of

One special case is when , so that there is only one cyclic prime-power factor in the Sylow -subgroup . In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when is arbitrary but for . Here, one is considering to be of the form

so elements of this subgroup can be viewed as comprising a vector space of dimension over the finite field of elements . The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

where is the appropriate general linear group. This is easily shown to have order

In the most general case, where the and are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines

and

then one has in particular , , and

One can check that this yields the orders in the previous examples as special cases (see Hillar, C., & Rhea, D.).

An abelian group A is finitely generated if it contains a finite set of elements (called *generators*) such that every element of the group is a linear combination with integer coefficients of elements of G.

Let L be a free abelian group with basis There is a unique group homomorphism such that

This homomorphism is surjective, and its kernel is finitely generated (since integers form a Noetherian ring). Consider the matrix M with integer entries, such that the entries of its jth column are the coefficients of the jth generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by M. Conversely every integer matrix defines a finitely generated abelian group.

It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices. In particular, changing the generating set of A is equivalent with multiplying M on the left by a unimodular matrix (that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of M is equivalent with multiplying M on the right by a unimodular matrix.

The Smith normal form of M is a matrix

where U and V are unimodular, and S is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries are the first ones, and is a divisor of for *i* > *j*. The existence and the shape of the Smith normal proves that the finitely generated abelian group A is the direct sum

where r is the number of zero rows at the bottom of r (and also the rank of the group). This is the fundamental theorem of finitely generated abelian groups.

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.

The simplest infinite abelian group is the infinite cyclic group . Any finitely generated abelian group is isomorphic to the direct sum of copies of and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of prime power orders. Even though the decomposition is not unique, the number , called the rank of , 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. Divisible groups, i.e. abelian groups in which the equation admits a solution for any natural number and element of , constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to and Prüfer groups for various prime numbers , and the cardinality of the set of summands of each type is uniquely determined.^{ [10] } Moreover, if a divisible group is a subgroup of an abelian group then admits a direct complement: a subgroup of such that . Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is called **reduced**.

Two important special classes of infinite abelian groups with diametrically opposite properties are *torsion groups* and *torsion-free groups*, exemplified by the groups (periodic) and (torsion-free).

An abelian group is called ** periodic ** or ** torsion **, if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Prüfer theorems state that if is a periodic group, and it either has a **bounded exponent**, i.e., for some natural number , or is countable and the -heights of the elements of are finite for each , then is isomorphic to a direct sum of finite cyclic groups.^{ [11] } The cardinality of the set of direct summands isomorphic to in such a decomposition is an invariant of .^{ [12] }^{:6} These theorems were later subsumed in the **Kulikov criterion**. 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.

An abelian group is called **torsion-free** if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively:

- Free abelian groups, i.e. arbitrary direct sums of
- Cotorsion and algebraically compact torsion-free groups such as the -adic integers
- Slender groups

An abelian group that is neither periodic nor torsion-free is called **mixed**. If is an abelian group and is its torsion subgroup, then the factor group is torsion-free. However, in general the torsion subgroup is not a direct summand of , so is *not* isomorphic to . Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group of integers is torsion-free -module.^{ [13] }^{:206}

One of the most basic invariants of an infinite abelian group is its rank: the cardinality of the maximal linearly independent subset of . Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of and can be completely described. More generally, a torsion-free abelian group of finite rank is a subgroup of . On the other hand, the group of -adic integers is a torsion-free abelian group of infinite -rank and the groups with different are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.

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. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. 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.

The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are:

- Tensor product
- Corner's results on countable torsion-free groups
- Shelah's work to remove cardinality restrictions.

Many large abelian groups possess a natural topology, which turns them into topological groups.

The collection of all abelian groups, together with the homomorphisms between them, forms the category , the prototype of an abelian category.

WandaSzmielew ( 1955 ) proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Most algebraic structures other than Boolean algebras are undecidable.

There are still many areas of current research:

- Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood;
- There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
- While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature.
- Many mild extensions of the first-order theory of abelian groups are known to be undecidable.
- Finite abelian groups remain a topic of research in computational group theory.

Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is:

- Undecidable in ZFC (Zermelo–Fraenkel axioms), the conventional axiomatic set theory from which nearly all of present-day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
- Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom;
- Positively answered if ZFC is augmented with the axiom of constructibility (see statements true in L).

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 acknowledgement 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.^{ [14] }

- Commutator subgroup – Smallest normal subgroup by which the quotient is commutative
- Abelianization – Quotienting a group by its commutator subgroup
- Dihedral group of order 6 – Non-commutative group with 6 elements, the smallest non-abelian group
- Elementary abelian group – Commutative group in which all nonzero elements have the same order
- Pontryagin duality – Duality for locally compact abelian groups

- ↑ Jacobson (2009) p. 41
- ↑ Ramík, J.,
*Pairwise Comparisons Method: Theory and Applications in Decision Making*(Cham: Springer Nature Switzerland, 2020), p. 11. - ↑ Auslander, M., & Buchsbaum, D.,
*Groups, Rings, Modules*(Mineola, NY: Dover Publications, 1974), pp. 28–29. - ↑ Rose 2012, p. 32.
- ↑ Cox, D. A.,
*Galois Theory*(Hoboken: John Wiley & Sons, 2004), pp. 144–145. - ↑ Dixon, M. R., Kurdachenko, L. A., & Subbotin, I. Y.,
*Linear Groups: The Accent on Infinite Dimensionality*(Milton Park, Abingdon-on-Thames & Oxfordshire: Taylor & Francis, 2020), pp. 49–50. - ↑ Rose 2012, p. 48.
- ↑ Rose 2012, p. 79.
- ↑ Kurzweil, H., & Stellmacher, B.,
*The Theory of Finite Groups: An Introduction*(New York, Berlin, Heidelberg: Springer Verlag, 2004), pp. 43–54. - ↑ For example, .
- ↑ Countability assumption in the second Prüfer theorem cannot be removed: the torsion subgroup of the direct product of the cyclic groups for all natural is not a direct sum of cyclic groups.
- ↑ Faith, C. C.,
*Rings and Things and a Fine Array of Twentieth Century Associative Algebra*(Providence: American Mathematical Society, 2004), p. 6. - ↑ Lal, R.,
*Algebra 2: Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier*(Berlin, Heidelberg: Springer, 2017), p. 206. - ↑ "Abel Prize Awarded: The Mathematicians' Nobel". Archived from the original on 31 December 2012. Retrieved 3 July 2016.

In mathematics, in the area of abstract algebra known as Galois theory, the **Galois group** of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

In mathematics, specifically in group theory, the concept of a **semidirect product** is a generalization of a direct product. There are two closely related concepts of semidirect product:

In group theory, a branch of abstract algebra, a **cyclic group** or **monogenous group** is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element *g* such that every other element of the group may be obtained by repeatedly applying the group operation to *g* or its inverse. Each element can be written as a power of *g* in multiplicative notation, or as a multiple of *g* in additive notation. This element *g* is called a *generator* of the group.

In abstract algebra, an abelian group is called **finitely generated** if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a *generating set* of or that *generate*.

In abstract algebra, the **direct sum** is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.

In mathematics, more specifically in the field of group theory, a **solvable group** or **soluble group** is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

An **exact sequence** is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.

In abstract algebra, a **Dedekind domain** or **Dedekind ring**, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

In mathematics, a **free abelian group** or **free Z-module** is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

In algebraic topology, the **Betti numbers** are used to distinguish topological spaces based on the connectivity of *n*-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces, the sequence of Betti numbers is 0 from some point onward, and they are all finite.

In algebra, a **group ring** is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is one-to-one with the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

In mathematics, especially in the field of group theory, a **divisible group** is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an *n*th multiple for each positive integer *n*. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups.

In mathematics, the **Ext functors** are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext^{1} classifies extensions of one module by another.

In mathematics, the **Grothendieck group** construction constructs an abelian group from a commutative monoid *M* in the most universal way, in the sense that any abelian group containing a homomorphic image of *M* will also contain a homomorphic image of the Grothendieck group of *M*. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

The **direct sum** is an operation from abstract algebra, a branch of mathematics. For example, the direct sum , where is real coordinate space, is the Cartesian plane, . To see how the direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of two vector spaces or two modules.

In mathematics, an **Azumaya algebra** is a generalization of central simple algebras to *R*-algebras where *R* need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where *R* is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964–65. There are now several points of access to the basic definitions.

In mathematics, specifically in group theory, the **direct product** is an operation that takes two groups *G* and *H* and constructs a new group, usually denoted *G* × *H*. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

In arithmetic geometry, the **Mordell–Weil group** is an abelian group associated to any abelian variety defined over a number field is an arithmetic invariant of the Abelian variety. It is simply the group of -points of , so is the Mordell–Weil group^{pg 207}. The main structure theorem about this group is the Mordell–Weil theorem which shows this group in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of to the zero of the associated L-function at a special point.

In mathematics, in the field of abstract algebra, the **structure theorem for finitely generated modules over a principal ideal domain** is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

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*Infinite Abelian Groups*. Pure and Applied Mathematics.**36-I**. Academic Press. MR 0255673. - Fuchs, László (1973).
*Infinite Abelian Groups*. Pure and Applied Mathematics. 36-II. Academic Press. MR 0349869. - Griffith, Phillip A. (1970).
*Infinite Abelian group theory*. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7. - Herstein, I. N. (1975).
*Topics in Algebra*(2nd ed.). John Wiley & Sons. ISBN 0-471-02371-X. - Hillar, Christopher; Rhea, Darren (2007). "Automorphisms of finite abelian groups".
*American Mathematical Monthly*.**114**(10): 917–923. arXiv: math/0605185 . Bibcode:2006math......5185H. doi:10.1080/00029890.2007.11920485. JSTOR 27642365. S2CID 1038507. - Jacobson, Nathan (2009).
*Basic Algebra I*(2nd ed.). Dover Publications. ISBN 978-0-486-47189-1. - Rose, John S. (2012).
*A Course on Group Theory*. Dover Publications. ISBN 978-0-486-68194-8. Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978. - Szmielew, Wanda (1955). "Elementary properties of abelian groups" (PDF).
*Fundamenta Mathematicae*.**41**(2): 203–271. doi: 10.4064/fm-41-2-203-271 . MR 0072131. Zbl 0248.02049.

- "Abelian group".
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