In
mathematics, a free abelian group is an
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 comm ...
with a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
. Being an abelian group means that it is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
with an addition
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
that is
associative,
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, and invertible. A basis, also called an integral basis, is a
subset such that every element of the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
can be uniquely expressed as an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
combination
In mathematics, a combination is a selection of items from a set that has distinct members, 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 th ...
of finitely many basis elements. For instance the two-dimensional
integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...
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 and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, and may equivalently be called free the
free modules 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 pair of elements has a unique supremum (also called a least upper bou ...
studies free abelian
subgroup
In group theory, a branch of mathematics, given a 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, ''H'' is a subgroup ...
s of
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
vector spaces. In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, free abelian groups are used to define
chain groups, and in
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 or evenly divisible by ...
.
The elements of a free abelian group with basis
may be described in several equivalent ways. These include formal sums which are expressions of the form
where each
is a nonzero integer, each
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, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
from
to the integers with finitely many nonzero values; for this functional representation, the group operation is the
pointwise In 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 ...
addition of functions.
Every set
has a free abelian group with
as its basis. This group is unique in the sense that every two free abelian groups with the same basis are
isomorphic. Instead of constructing it by describing its individual elements, a free abelian group with basis
may be constructed as a
direct sum of copies of the additive group of the integers, with one copy per member Alternatively, the free abelian group with basis
may be described by a
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
with the elements of
as its generators and with the
commutators 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, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of a free abelian group is the
cardinality 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, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
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 ...
of an
injective homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
between free abelian groups. The only free abelian groups that are free groups are the
trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
and the
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
.
Definition and examples
![Lattice in R2](https://upload.wikimedia.org/wikipedia/commons/e/ed/Lattice_in_R2.svg)
A free abelian group is an
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 comm ...
that has a basis. Here, being an abelian group means that it is described by a set
of its elements and a
binary operation 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 structures ...
by the
symbol (although it need not be the usual addition of numbers) that obey the following properties:
*The operation
is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
and
associative, meaning for all elements
and Therefore, when combining two or more elements of
using this operation, the ordering and grouping of the elements does not affect the result.
*
contains an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
(conventionally denoted with the property that, for every
*Every element
in
has an
inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
such that
A basis is a subset
of the elements of
with the property that every element of
may be formed in a unique way by choosing finitely many basis elements
choosing a nonzero integer
for each of the chosen basis elements, and adding together
copies of the basis elements
for which
is positive, and
copies of
for each basis element for which
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, an empty sum, or nullary sum, is a summation where the number of terms is zero.
The natural way to extend non-empty sums is to let the empty sum be the additive identity.
Let a_1, a_2, a_3, ... be a sequence of numbers, and let
...
, and it must not be possible to find any other combination that represents the identity.
[Some sources define free abelian groups by the condition that the only representation of the identity is the empty sum, rather than treating it as a special case of unique representation of all group elements; see, e.g., .]
The under the usual addition operation, form a free abelian group with the The integers are commutative and associative, with
0 as the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elemen ...
and with each integer having an
additive inverse, its negation. Each non-negative
is the sum of
copies and each negative integer
is the sum of
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s which form a free abelian group with the usual
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 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 because the only ways ...
s as their basis. Multiplication is commutative and associative, with the number
as its identity and with
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 mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
, according to which every positive integer can be
factorized uniquely into the product of finitely many primes or their inverses. If
is a positive rational number expressed in simplest terms, then
can be expressed as a finite combination of the primes appearing in the factorizations of
The number of copies of each prime to use in this combination is its exponent in the factorization of
, or the negation of its exponent in the factorization
The
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s of a single with integer coefficients, form a free abelian group under polynomial addition, with the powers of
as a basis. As an abstract group, this is the same as (an
isomorphic group 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
in the corresponding polynomial, or vice versa. For instance the rational number
has exponents of
for the first three prime numbers
and would correspond in this way to the polynomial
having the same coefficients
for its constant, linear, and quadratic terms. Because these mappings merely reinterpret the same numbers, they define a
bijection between the elements of the two groups. And because the group operation of multiplying positive rationals acts additively on the exponents of the prime numbers, in the same way that the group operation of adding polynomials acts on the coefficients of the polynomials, these maps preserve the group structure; they are
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s. A bijective homomorphism is called an isomorphism, and its existence demonstrates that these two groups have the same properties.
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 inverse, one gets another basis. As a more elaborated example, the two-dimensional
integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...
consisting of the points in the plane with integer
Cartesian coordinates, forms a free abelian group under
vector addition
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
with the basis For this basis, the element
can be written where 'multiplication' is defined so that, for instance, There is no other way to write
in the same basis. However, with a different basis such as it can be written as Generalizing this example, every
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
forms a
finitely-generated free abelian group. The integer lattice
has a natural basis consisting of the positive integer
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
s, but it has many other bases as well: if
is a
integer
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with then the rows of
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 In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined.
Definition ...
.
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 a
presentation of a group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
.
Products and sums
The
direct product of groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one ...
consists of tuples of an element from each group in the product, with componentwise addition. The direct product of two free abelian groups is itself free abelian, with basis the
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
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
copies of the integer The trivial group
is also considered to be free abelian, with basis the
empty set. It may be interpreted as an
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
, 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, in the field of group theory, the Baer–Specker group, or Specker group, named after Reinhold Baer and Ernst Specker, is an example of an infinite Abelian group which is a building block in the structure theory of such groups.
Def ...
an
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
group formed as the direct product of
countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
many copies was shown in 1937 by
Reinhold Baer
Reinhold Baer (22 July 1902 – 22 October 1979) was a 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 engineering f ...
to not be free abelian, although
Ernst Specker
Ernst Paul Specker (11 February 1920, Zurich – 10 December 2011, Zurich) was a Swiss mathematician. Much of his most influential work was on Quine's New Foundations, a set theory with a universal set, but he is most famous for the Kochen ...
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 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
to become the basis of a free abelian group.
Integer functions and formal sums
Given a one can define a group
whose elements are functions from
to the integers, where the parenthesis in the superscript indicates that only the functions with finitely many nonzero values are included.
If
and
are two such functions, then
is the function whose values are sums of the values in
that is, This
pointwise In 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 ...
addition operation gives
the structure of an abelian group.
Each element
from the given set
corresponds to a member the function
for which
and for which
for
Every function
in
is uniquely a linear combination of a finite number of basis elements:
Thus, these elements
form a basis and
is a free abelian group.
In this way, every set
can be made into the basis of a free abelian group.
The elements of
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, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
s of finitely many elements
Presentation
A
presentation of a group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
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 elements of a group defined in this way are
equivalence classes of sequences of generators and their inverses, under an
equivalence relation that allows inserting or removing any relator or generator-inverse pair as a contiguous subsequence. The free abelian group with basis
has a presentation in which the generators are the elements and the relators are the
commutators of pairs of elements Here, the commutator of two elements
and
is the product setting this product to the identity causes
to so that
and
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) can be used to show that every finitely generated abelian group is finitely presented. For, if
is finitely generated by a it is a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the free abelian group over
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
* Mo ...
s over the integers are defined similarly to
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
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 in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
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 "free" for those that do. A
free module is a module that can be represented as a direct sum over its base
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, 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, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of The tensor product of two free abelian groups is always free abelian, with a basis that is the
Cartesian product 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. For instance,
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
s of free modules over principal ideal domains are free, a fact that writes allows for "automatic generalization" of
homological
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
machinery to these modules. Additionally, the theorem that every
projective is free generalizes in the same way.
Properties
Universal property
A free abelian group
with basis
has the following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
: for every function
from
to an abelian group
, there exists a unique
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
from
to
which extends
. Here, a group homomorphism is a mapping from one group to the other that is consistent with the group product law: performing a product before or after the mapping produces the same result. By a general property of universal properties, this shows that "the" abelian group of base
is unique
up to an isomorphism. Therefore, the universal property can be used as a definition of the free abelian group of base
. 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, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between eleme ...
s in the
category of abelian groups, the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
that has abelian groups as its objects and homomorphisms as its arrows. The map from a basis to its free abelian group is a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
, a structure-preserving mapping of categories, from sets to abelian groups, and is
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
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 sign ...
from abelian groups to sets. However, a ''free abelian'' group is ''not'' a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
except in two cases: a free abelian group having an empty basis (rank zero, giving the
trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
) or having just one element in the basis (rank one, giving the
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
). Other abelian groups are not free groups because in free groups
must be different from
if
and
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, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
There a ...
, it is an added constraint to demand that
, 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, so the cardinality of a basis forms an
invariant of the group known as its 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
, in which case the group is isomorphic to
.
This notion of rank can be generalized, from free abelian groups to abelian groups that are not necessarily free. The
rank of an abelian group is defined as the rank of a free abelian subgroup
of
for which the
quotient group is a
torsion group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements.
For examp ...
. Equivalently, it is the cardinality of a
maximal subset of
that generates a free subgroup. The rank 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 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 and Otto Schreier.
Statement of the theorem
A free group may be defined from a grou ...
that every subgroup of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. A proof using
Zorn's lemma (one of many equivalent assumptions to the axiom of choice) can be found in
Serge Lang's ''Algebra''.
Solomon Lefschetz
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
and
Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...
argue 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 i ...
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
is a subgroup of a finitely generated free abelian group
, then
is free and there exists a basis
of
and positive integers
(that is, each one divides the next one) such that
is a basis of
Moreover, the sequence
depends only on
and
and not on the basis. A
constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existenc ...
of the existence part of the theorem is provided by any algorithm computing the
Smith normal form In 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, and can b ...
of a matrix of integers. Uniqueness follows from the fact that, for any
, 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 ...
of the
minors of rank
of the matrix is not changed during the Smith normal form computation and is the product
at the end of the computation.
Torsion and divisibility
All free abelian groups are
torsion-free, meaning that there is no non-identity group element
and nonzero integer
such that
.
Conversely, all finitely generated torsion-free abelian groups are free abelian.
The additive group of rational numbers
provides an example of a torsion-free (but not finitely generated) abelian group that is not free abelian. One reason that
is not free abelian is that it is
divisible
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 or evenly divisible by ...
, meaning that, for every element
and every nonzero integer
, it is possible to express
as a scalar multiple
of another element
. 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 automorphism
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 grou ...
s, the
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
homomorphisms from the group to itself. In non-abelian groups these are further subdivided into
inner
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
and
outer
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