Abelian group

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.
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.
Definition.
An abelian group is a set, formula_1, together with an operation formula_2 that combines any two elements formula_3 and formula_4 of formula_1 to form another element of formula_6 denoted formula_7. The symbol formula_2 is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, formula_9, must satisfy five requirements known as the "abelian group axioms":
A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".
Facts.
Notation.
There are two main notational conventions for abelian groups – additive and multiplicative.
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.
Multiplication table.
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 formula_35 under the the entry of this table contains the product formula_36.
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 formula_37 for all formula_38, which is iff the formula_39 entry of the table equals the formula_40 entry for all formula_38, i.e. the table is symmetric about the main diagonal.
Examples.
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 formula_60 rotation matrices.
Historical remarks.
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.
Properties.
If formula_44 is a natural number and formula_50 is an element of an abelian group formula_49 written additively, then formula_64 can be defined as formula_65 (formula_44 summands) and formula_67. In this way, formula_49 becomes a module over the ring formula_54 of integers. In fact, the modules over formula_54 can be identified with the abelian groups.
Theorems about abelian groups (i.e. modules over the principal ideal domain formula_54) 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 formula_72 for formula_73 prime, and the latter is a direct sum of finitely many copies of formula_54.
If formula_75 are two group homomorphisms between abelian groups, then their sum formula_76, defined by formula_77, is again a homomorphism. (This is not true if formula_78 is a non-abelian group.) The set formula_79 of all group homomorphisms from formula_49 to formula_78 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. 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 formula_82 of a group formula_49 is the set of elements that commute with every element of formula_49. A group formula_49 is abelian if and only if it is equal to its center formula_82. The center of a group formula_49 is always a characteristic abelian subgroup of formula_49. If the quotient group formula_89 of a group by its center is cyclic then formula_49 is abelian.
Finite abelian groups.
Cyclic groups of integers modulo formula_44, formula_92, 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. In fact, for every prime number formula_73 there are (up to isomorphism) exactly two groups of order formula_94, namely formula_95 and formula_96.
Classification.
The fundamental theorem of finite abelian groups states that every finite abelian group formula_49 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. 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 formula_98 of order formula_99 is isomorphic to the direct sum of formula_100 and formula_101 if and only if formula_47 and formula_44 are coprime. It follows that any finite abelian group formula_49 is isomorphic to a direct sum of the form
in either of the following canonical ways:
For example, formula_111 can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: formula_112. 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 formula_113 (the integers 0 to 7 under addition modulo 8), formula_114 (the odd integers 1 to 15 under multiplication modulo 16), or formula_115.
See also list of small groups for finite abelian groups of order 30 or less.
Automorphisms.
One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group formula_49. To do this, one uses the fact that if formula_49 splits as a direct sum formula_118 of subgroups of coprime order, then
Given this, the fundamental theorem shows that to compute the automorphism group of formula_49 it suffices to compute the automorphism groups of the Sylow formula_73-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of formula_73). Fix a prime formula_73 and suppose the exponents formula_124 of the cyclic factors of the Sylow formula_73-subgroup are arranged in increasing order:
for some formula_127. One needs to find the automorphisms of
One special case is when formula_129, so that there is only one cyclic prime-power factor in the Sylow formula_73-subgroup formula_131. In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when formula_44 is arbitrary but formula_133 for formula_134. Here, one is considering formula_131 to be of the form
so elements of this subgroup can be viewed as comprising a vector space of dimension formula_44 over the finite field of formula_73 elements formula_139. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so
where formula_141 is the appropriate general linear group. This is easily shown to have order
In the most general case, where the formula_124 and formula_44 are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines
and
then one has in particular formula_147, formula_148, and
One can check that this yields the orders in the previous examples as special cases (see Hillar, C., & Rhea, D.).
Finitely generated abelian groups.
An abelian group is finitely generated if it contains a finite set of elements (called "generators") formula_150 such that every element of the group is a linear combination with integer coefficients of elements of . 
Let be a free abelian group with basis formula_151 
There is a unique group homomorphism
formula_152 such that 
This homomorphism is surjective, and its kernel is finitely generated (since integers form a Noetherian ring). Consider the matrix with integer entries, such that the entries of its th column are the coefficients of the th generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by . 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 is equivalent with multiplying 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 is equivalent with multiplying on the right by a unimodular matrix.
The Smith normal form of is a matrix
where and are unimodular, and 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 . The existence and the shape of the Smith normal proves that the finitely generated abelian group is the direct sum 
where is the number of zero rows at the bottom of (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.
Infinite abelian groups.
The simplest infinite abelian group is the infinite cyclic group formula_54. Any finitely generated abelian group formula_1 is isomorphic to the direct sum of formula_158 copies of formula_54 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 formula_158, called the rank of formula_1, 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 formula_1 in which the equation formula_163 admits a solution formula_164 for any natural number formula_44 and element formula_3 of formula_1, 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 formula_168 and Prüfer groups formula_169 for various prime numbers formula_73, and the cardinality of the set of summands of each type is uniquely determined. Moreover, if a divisible group formula_1 is a subgroup of an abelian group formula_49 then formula_1 admits a direct complement: a subgroup formula_174 of formula_49 such that formula_176. 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 formula_177 (periodic) and formula_168 (torsion-free).
Torsion groups.
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 formula_1 is a periodic group, and it either has a bounded exponent, i.e., formula_180 for some natural number formula_44, or is countable and the formula_73-heights of the elements of formula_1 are finite for each formula_73, then formula_1 is isomorphic to a direct sum of finite cyclic groups. 
The cardinality of the set of direct summands isomorphic to formula_186 in such a decomposition is an invariant of formula_1. 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 formula_73-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.
Torsion-free and mixed groups.
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:
An abelian group that is neither periodic nor torsion-free is called mixed. If formula_1 is an abelian group and formula_192 is its torsion subgroup, then the factor group formula_193 is torsion-free. However, in general the torsion subgroup is not a direct summand of formula_1, so formula_1 is "not" isomorphic to formula_196. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group formula_54 of integers is torsion-free formula_54-module.
Invariants and classification.
One of the most basic invariants of an infinite abelian group formula_1 is its rank: the cardinality of the maximal linearly independent subset of formula_1. Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of formula_168 and can be completely described. More generally, a torsion-free abelian group of finite rank formula_158 is a subgroup of formula_203. On the other hand, the group of formula_73-adic integers formula_205 is a torsion-free abelian group of infinite formula_54-rank and the groups formula_207 with different formula_44 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.
Additive groups of rings.
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:
Relation to other mathematical topics.
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 formula_209, the prototype of an abelian category.
 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:
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:
A note on the typography.
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.