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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a free abelian group is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
with a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...

basis
. Being an abelian group means that it is a set with an addition operation that is
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
combination In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of t ...
of finitely many basis elements. For instance the two-dimensional
integer latticeIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s, and may equivalently be called free the
free module In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

free module
s over the integers.
Lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper boun ...
studies free abelian
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
s of real vector spaces. In
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
, free abelian groups are used to define chain groups, and in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
they are used to define
divisors In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m ...
. The elements of a free abelian group with basis B may be described in several equivalent ways. These include formal sums which are expressions of the form \sum a_i b_i where each a_i is a nonzero integer, each b_i is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed
multiset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s containing finitely many elements with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from B to the integers with finitely many nonzero values; for this functional representation, the group operation is the
pointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...
addition of functions. Every set B has a free abelian group with B as its basis. This group is unique in the sense that every two free abelian groups with the same basis are
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. Instead of constructing it by describing its individual elements, a
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
with basis B may be constructed as a
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of copies of the additive group of the integers, with one copy per member Alternatively, the free abelian group with basis B may be described by a
presentation A presentation conveys information from a speaker to an audience An audience is a group of people who participate in a show or encounter a work of art A work of art, artwork, art piece, piece of art or art object is an a ...
with the elements of B as its generators and with the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s of pairs of members as its relators. The
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...
of a free abelian group is the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
of a free abelian group by "relations", or as a
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...

cokernel
of an injective
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

homomorphism
between free abelian groups. The only free abelian groups that are free groups are the
trivial groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
and the
infinite cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...
.


Definition and examples

A free abelian group is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
that has a basis. Here, being an abelian group means that it is described by a set S of its elements and a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
conventionally denoted as an
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
by the + symbol (although it need not be the usual addition of numbers) that obey the following properties: *The operation + is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
and
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, meaning for all elements x+y=y+x and Therefore, when combining two or more elements of S using this operation, the ordering and grouping of the elements does not affect the result. *S contains an
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
(conventionally denoted with the property that, for every *Every element x in S has an
inverse element In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
such that A basis is a subset B of the elements of S with the property that every element of S may be formed in a unique way by choosing finitely many basis elements b_i choosing a nonzero integer k_i for each of the chosen basis elements, and adding together k_i copies of the basis elements b_i for which k_i is positive, and -k_i copies of -b_i for each basis element for which k_i is negative. As a special case, the identity element can always be formed in this way as the combination of zero basis elements, according to the usual convention for an
empty sum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and it must not be possible to find any other combination that represents the identity. The under the usual addition operation, form a free abelian group with the The integers are commutative and associative, with
0
0
as the
additive identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
and with each integer having an
additive inverse In mathematics, the additive inverse of a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be ...
, its negation. Each non-negative x is the sum of x copies and each negative integer x is the sum of -x copies so the basis property is also satisfied. An example where the group operation is different from the usual addition of numbers is given by the positive
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s which form a free abelian group with the usual
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
operation on numbers and with the
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s as their basis. Multiplication is commutative and associative, with the number 1 as its identity and with 1/x as the inverse element for each positive rational The fact that the prime numbers forms a basis for multiplication of these numbers follows from the
fundamental theorem of arithmetic In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
, according to which every positive integer can be factorized uniquely into the product of finitely many primes or their inverses. If q=a/b is a positive rational number expressed in simplest terms, then q can be expressed as a finite combination of the primes appearing in the factorizations of a The number of copies of each prime to use in this combination is its exponent in the factorization of a, or the negation of its exponent in the factorization The
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s of a single with integer coefficients, form a free abelian group under polynomial addition, with the powers of x as a basis. As an abstract group, this is the same as (an
isomorphic group In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two gr ...
to) the multiplicative group of positive rational numbers. One way to map these two groups to each other, showing that they are isomorphic, is to reinterpret the exponent of the prime number in the multiplicative group of the rationals as instead giving the coefficient of x^ in the corresponding polynomial, or vice versa. Although the representation of each group element in terms of a given basis is unique, a free abelian group has generally more than one basis, and different bases will generally result in different representations of its elements. For example, if one replaces any element of a basis by its negation, one gets another basis. As a more elaborated example, the two-dimensional
integer latticeIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
consisting of the points in the plane with integer
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Cartesian coordinates
, forms a free abelian group under
vector addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

vector addition
with the basis For this basis, the element (4,3) can be written where 'multiplication' is defined so that, for instance, There is no other way to write (4,3) in the same basis. However, with a different basis such as it can be written as Generalizing this example, every lattice forms a finitely-generated free abelian group. The integer lattice \Z^d has a natural basis consisting of the positive integer
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s, but it has many other bases as well: if M is a d\times d integer matrix with then the rows of M form a basis, and conversely every basis of the integer lattice has this form. For more on the two-dimensional case, see fundamental pair of periods.


Constructions

Every set can be the basis of a free abelian group, which is unique up to group isomorphisms. The free abelian group for a given basis set can be constructed in several different but equivalent ways: as a direct sum of copies of the integers, as a family of integer-valued functions, as a signed multiset, or by
presentation of a group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
.


Products and sums

The
direct product of groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
consists of tuples of an element from each group in the product, with pointwise addition. The direct product of two free abelian groups is itself free abelian, with basis the
disjoint union In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

disjoint union
of the bases of the two groups. More generally the direct product of any finite number of free abelian groups is free abelian. The integer lattice, for instance, is isomorphic to the direct product of d copies of the integer The trivial group \ is also considered to be free abelian, with basis the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

empty set
. It may be interpreted as an
empty product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, the direct product of zero copies For infinite families of free abelian groups, the direct product is not necessarily free abelian. For instance the
Baer–Specker group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
an uncountable group formed as the direct product of
countably In mathematics, a countable set is a Set (mathematics), set with the same cardinality (cardinal number, number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether f ...
many copies was shown in 1937 by
Reinhold Baer Reinhold Baer (22 July 1902 – 22 October 1979) was a Germany, German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings and Baer groups. Biography Baer studied mechanical engine ...
to not be free abelian, although
Ernst Specker Ernst Paul Specker (11 February 1920, Zurich – 10 December 2011, Zurich) was a Switzerland, Swiss mathematician. Much of his most influential work was on Willard Van Orman Quine, Quine's New Foundations, a set theory with a universal set, but he ...
proved in 1950 that all of its countable subgroups are free abelian. Instead, to obtain a free abelian group from an infinite family of groups, the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
rather than the direct product should be used. The direct sum and direct product are the same when they are applied to finitely many groups, but differ on infinite families of groups. In the direct sum, the elements are again tuples of elements from each group, but with the restriction that all but finitely many of these elements are the identity for their group. The direct sum of infinitely many free abelian groups remains free abelian. It has a basis consisting of tuples in which all but one element is the identity, with the remaining element part of a basis for its group. Every free abelian group may be described as a direct sum of copies with one copy for each member of its basis. This construction allows any set B to become the basis of a free abelian group.


Integer functions and formal sums

Given a one can define a group \mathbb^ whose elements are functions from B to the integers, where the parenthesis in the superscript indicates that only the functions with finitely many nonzero values are included. If f(x) and g(x) are two such functions, then f+g is the function whose values are sums of the values in f that is, This
pointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...
addition operation gives \mathbb^ the structure of an abelian group. Each element x from the given set B corresponds to a member the function e_x for which e_x(x)=1 and for which e_x(y)=0 for Every function f in \mathbb^ is uniquely a linear combination of a finite number of basis elements: f=\sum_ f(x) e_x. Thus, these elements e_x form a basis and \mathbb^ is a free abelian group. In this way, every set B can be made into the basis of a free abelian group. The elements of \mathbb^ may also be written as formal sums, expressions in the form of a sum of finitely many terms, where each term is written as the product of a nonzero integer with a distinct member These expressions are considered equivalent when they have the same terms, regardless of the ordering of terms, and they may be added by forming the union of the terms, adding the integer coefficients to combine terms with the same basis element, and removing terms for which this combination produces a zero coefficient. They may also be interpreted as the signed
multiset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s of finitely many elements


Presentation

A
presentation of a group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
is a set of elements that generate the group (meaning that all group elements can be expressed as products of finitely many generators), together with "relators", products of generators that give the identity element. The free abelian group with basis B has a presentation in which the generators are the elements and the relators are the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s of pairs of elements Here, the commutator of two elements x and y is the product setting this product to the identity causes xy to so that x and y commute. More generally, if all pairs of generators commute, then all pairs of products of generators also commute. Therefore, the group generated by this presentation is abelian, and the relators of the presentation form a minimal set of relators needed to ensure that it is abelian. When the set of generators is finite, the presentation of a free abelian group is also finite, because there are only finitely many different commutators to include in the presentation. This fact, together with the fact that every subgroup of a free abelian group is free abelian (
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
) can be used to show that every finitely generated abelian group is finitely presented. For, if G is finitely generated by a it is a quotient of the free abelian group over B by a free abelian subgroup, the subgroup generated by the relators of the presentation But since this subgroup is itself free abelian, it is also finitely generated, and its basis (together with the commutators forms a finite set of relators for a presentation


As a module

The
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
s over the integers are defined similarly to
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s over the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s or
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s: they consist of systems of elements that can be added to each other, with an operation for
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module (mathematics), module in abstract algebra). In common geometrical contexts, scalar multiplication of a re ...

scalar multiplication
by integers that is compatible with this addition operation. Every abelian group may be considered as a module over the integers, with a scalar multiplication operation defined as follows: However, unlike vector spaces, not all abelian groups have a basis, hence the special name for those that do. A
free module In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

free module
is a module that can be represented as a direct sum over its base ring, so free abelian groups and free are equivalent concepts: each free abelian group is (with the multiplication operation above) a free and each free comes from a free abelian group in this way. As well as the direct sum, another way to combine free abelian groups is to use the
tensor product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

tensor product
of The tensor product of two free abelian groups is always free abelian, with a basis that is the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of the bases for the two groups in the product. Many important properties of free abelian groups may be generalized to free modules over a
principal ideal domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. For instance, submodules of free modules over principal ideal domains are free, a fact that writes allows for "automatic generalization" of homological machinery to these modules. Additionally, the theorem that every projective is free generalizes in the same way.


Properties


Universal property

A free abelian group F with basis B has the following
universal property In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
: for every function f from B to an abelian group A, there exists a unique
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

group homomorphism
from F to A which extends f. By a general property of universal properties, this shows that "the" abelian group of base B is unique
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
an isomorphism. Therefore, the universal property can be used as a definition of the free abelian group of base B. The uniqueness of the group defined by this property shows that all the other definitions are equivalent. It is because of this universal property that free abelian groups are called "free": they are the
free object In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s in the
category of abelian groupsIn mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...
, and the map from a basis to its free abelian group is a
functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

functor
from sets to abelian groups, adjoint to the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...
from abelian groups to sets. However, a ''free abelian'' group is ''not'' a
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
except in two cases: a free abelian group having an empty basis (rank zero, giving the
trivial groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
) or having just one element in the basis (rank one, giving the
infinite cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...
). Other abelian groups are not free groups because in free groups ab must be different from ba if a and b are different elements of the basis, while in free abelian groups the two products must be identical for all pairs of elements. In the general
category of groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, it is an added constraint to demand that ab=ba, whereas this is a necessary property in the category of abelian groups.


Rank

Every two bases of the same free abelian group have the same
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, so the cardinality of a basis forms an invariant of the group known as its
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...
. Two free abelian groups are isomorphic if and only if they have the same rank. A free abelian group is finitely generated if and only if its rank is a finite number n, in which case the group is isomorphic to \mathbb^n. This notion of rank can be generalized, from free abelian groups to abelian groups that are not necessarily free. The
rank of an abelian groupIn mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group ''A'' is the cardinality of a maximal linearly independent subset. The rank of ''A'' determines the size of the largest free abelian group contained in ''A''. If ''A'' i ...
G is defined as the rank of a free abelian subgroup F of G for which the
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
G/F is a
torsion group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by E ...
. Equivalently, it is the cardinality of a maximal subset of G that generates a free subgroup. Again, this is a group invariant; it does not depend on the choice of the subgroup.


Subgroups

Every subgroup of a free abelian group is itself a free abelian group. This result of
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
was a precursor to the analogous
Nielsen–Schreier theorem In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen (mathematician), Jakob Nielsen and Otto Schreier. Statement of the theorem A free gr ...
that every subgroup of a
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
is free, and is a generalization of the fact that every nontrivial subgroup of the infinite cyclic group is infinite cyclic. The proof needs the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
. A proof using
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...
(one of many equivalent assumptions to the axiom of choice) can be found in
Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

Serge Lang
's ''Algebra''.
Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gre ...
and
Irving Kaplansky Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (n ...
have claimed that using the
well-ordering principle In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which x precedes y if ...
in place of Zorn's lemma leads to a more intuitive proof. In the case of finitely generated free abelian groups, the proof is easier, does not need the axiom of choice, and leads to a more precise result. If G is a subgroup of a finitely generated free abelian group F, then G is free and there exists a basis (e_1, \ldots, e_n) of F and positive integers d_1, d_2, \ldots, d_k (that is, each one divides the next one) such that (d_1e_1,\ldots, d_ke_k) is a basis of G. Moreover, the sequence d_1,d_2,\ldots,d_k depends only on F and G and not on the particular basis (e_1, \ldots, e_n) that solves the problem. A
constructive proofIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of the existence part of the theorem is provided by any algorithm computing the
Smith normal formIn mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal The dia ...
of a matrix of integers. Uniqueness follows from the fact that, for any r\le k, the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

greatest common divisor
of the minors of rank r of the matrix is not changed during the Smith normal form computation and is the product d_1\cdots d_r at the end of the computation.


Torsion and divisibility

All free abelian groups are torsion-free, meaning that there is no group element (non-identity) x and nonzero integer n such that nx=0. Conversely, all finitely generated torsion-free abelian groups are free abelian. The additive group of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s \mathbb provides an example of a torsion-free (but not finitely generated) abelian group that is not free abelian. One reason that \mathbb is not free abelian is that it is
divisible In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, meaning that, for every element x\in\mathbb and every nonzero integer n, it is possible to express x as a scalar multiple ny of another element y=x/n. In contrast, non-trivial free abelian groups are never divisible, because in a free abelian group the basis elements cannot be expressed as multiples of other elements.


Symmetry

The symmetries of any group can be described as group automorphisms, the
invertible
invertible
homomorphisms from the group to itself. In non-abelian groups these are further subdivided into inner and outer automorphisms, but in abelian groups all non-identity automorphisms are outer. They form another group, the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
of the given group, under the operation of
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
. The automorphism group of a free abelian group of finite rank n is the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
GL(n,\mathbb), which can be described concretely (for a specific basis of the free automorphism group) as the set of n\times n invertible integer
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
under the operation of matrix multiplication. Their action as symmetries on the free abelian group \Z^n is just matrix-vector multiplication. The automorphism groups of two infinite-rank free abelian groups have the same first-order theories as each other, if and only if their ranks are equivalent cardinals from the point of view of
second-order logic In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
. This result depends on the structure of involutions of free abelian groups, the automorphisms that are their own inverse. Given a basis for a free abelian group, one can find involutions that map any set of disjoint pairs of basis elements to each other, or that negate any chosen subset of basis elements, leaving the other basis elements fixed. Conversely, for every involution of a free abelian group, one can find a basis of the group for which all basis elements are swapped in pairs, negated, or left unchanged by the involution.


Relation to other groups

If a free abelian group is a quotient of two groups A/B, then A is the direct sum B\oplus A/B. Given an arbitrary abelian group A, there always exists a free abelian group F and a
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
group homomorphism from F to A. One way of constructing a surjection onto a given group A is to let F=\mathbb^ be the free abelian group over A, represented as formal sums. Then a surjection can be defined by mapping formal sums in F to the corresponding sums of members of A. That is, the surjection maps \sum_ a_x e_x \mapsto \sum_ a_x x, where a_x is the integer coefficient of basis element e_x in a given formal sum, the first sum is in F, and the second sum is in A. This surjection is the unique group homomorphism which extends the function e_x\mapsto x, and so its construction can be seen as an instance of the universal property. When F and A are as above, the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
G of the surjection from F to A is also free abelian, as it is a subgroup of F (the subgroup of elements mapped to the identity). Therefore, these groups form a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics ...
0\to G\to F\to A\to 0 in which F and G are both free abelian and A is isomorphic to the
factor group A quotient group or factor group is a math Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...
F/G. This is a
free resolution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of A. Furthermore, assuming the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
, the free abelian groups are precisely the
projective objects
projective objects
in the
category of abelian groupsIn mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...
.


Applications


Algebraic topology

In
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
, a formal sum of k-dimensional
simplices
simplices
is called a k-chain, and the free abelian group having a collection of k-simplices as its basis is called a chain group. The simplices are generally taken from some topological space, for instance as the set of k-simplices in a
simplicial complex In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, or the set of
singular Singular may refer to: * Singular, the grammatical number In linguistics, grammatical number is a grammatical category of nouns, pronouns, adjectives, and verb agreement (linguistics), agreement that expresses count distinctions (such as "one", ...
k-simplices in a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
. Any k-dimensional simplex has a boundary that can be represented as a formal sum of (k-1)-dimensional simplices, and the universal property of free abelian groups allows this boundary operator to be extended to a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

group homomorphism
from k-chains to (k-1)-chains. The system of chain groups linked by boundary operators in this way forms a
chain complex In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and the study of chain complexes forms the basis of
homology theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.


Algebraic geometry and complex analysis

Every
rational function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

rational function
over the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s can be associated with a signed multiset of complex numbers c_i, the
zeros and poles In complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) incl ...
of the function (points where its value is zero or infinite). The multiplicity m_i of a point in this multiset is its order as a zero of the function, or the negation of its order as a pole. Then the function itself can be recovered from this data, up to a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
factor, as f(q)=\prod (q-c_i)^. If these multisets are interpreted as members of a free abelian group over the complex numbers, then the product or quotient of two rational functions corresponds to the sum or difference of two group members. Thus, the multiplicative group of rational functions can be factored into the multiplicative group of complex numbers (the associated scalar factors for each function) and the free abelian group over the complex numbers. The rational functions that have a nonzero limiting value at infinity (the
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function (mathematics), function that is holomorphic function, holomorphic on all of ''D'' ''except'' for a set of isolated p ...
s on the
Riemann sphere In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Riemann sphere
) form a subgroup of this group in which the sum of the multiplicities is zero. This construction has been generalized, in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
, to the notion of a
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. There are different definitions of divisors, but in general they form an abstraction of a codimension-one subvariety of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
, the set of solution points of a system of polynomial equations. In the case where the system of equations has one degree of freedom (its solutions form an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
or
Riemann surface In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
), a subvariety has codimension one when it consists of isolated points, and in this case a divisor is again a signed multiset of points from the variety. The meromorphic functions on a compact Riemann surface have finitely many zeros and poles, and their divisors can again be represented as elements of a free abelian group, with multiplication or division of functions corresponding to addition or subtraction of group elements. However, in this case there are additional constraints on the divisor beyond having zero sum of multiplicities.


Group rings

The
group ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
\Z /math>, for any group G, is ring whose additive group is the free abelian group over G. When G is finite and abelian, the multiplicative group of
units Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
in \Z /math> has the structure of a direct product of a finite group and a finitely generated free abelian group.


References

{{DEFAULTSORT:Free Abelian Group Abelian group theory Properties of groups Free algebraic structures