In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, 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 commut ...
is called finitely generated if there exist finitely many elements
in
such that every
in
can be written in the form
for some
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 language ...
s
. In this case, we say that the set
is a ''
generating set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
'' of
or that
''generate''
.
Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.
Examples
* The
integers
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 language o ...
,
, are a finitely generated abelian group.
* The
integers modulo ,
, are a finite (hence finitely generated) abelian group.
* Any
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of finitely many finitely generated abelian groups is again a finitely generated abelian group.
* 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 ornam ...
forms a finitely generated
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
.
There are no other examples (up to isomorphism). In particular, the group
of
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 ration ...
s is not finitely generated:
[Silverman & Tate (1992), p. 102/ref> if are rational numbers, pick a ]natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to all the denominators; then cannot be generated by . The group of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition and non-zero real numbers under multiplication are also not finitely generated.
Classification
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of ''finite'' abelian groups. The theorem, in both forms, in turn generalizes to the , which in turn admits further generalizations.
Primary decomposition
The primary decomposition formulation states that every finitely generated abelian group ''G'' is isomorphic to a direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of primary cyclic group
In mathematics, a primary cyclic group is a group that is both a cyclic group and a ''p''-primary group for some prime number ''p''.
That is, it is a cyclic group of order ''p'', C, for some prime number ''p'', and natural number ''m''.
Every f ...
s and 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 bina ...
s. A primary cyclic group is one whose order is a power of a prime
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 ...
. That is, every finitely generated abelian group is isomorphic to a group of the form
:
where ''n'' ≥ 0 is 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
* H ...
'', and the numbers ''q''1, ..., ''q''''t'' are powers of (not necessarily distinct) prime numbers. In particular, ''G'' is finite if and only if ''n'' = 0. The values of ''n'', ''q''1, ..., ''q''''t'' are (up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
rearranging the indices) uniquely determined by ''G'', that is, there is one and only one way to represent ''G'' as such a decomposition.
The proof of this statement uses the basis theorem for finite 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 commu ...
: every finite abelian group is a direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of primary cyclic group
In mathematics, a primary cyclic group is a group that is both a cyclic group and a ''p''-primary group for some prime number ''p''.
That is, it is a cyclic group of order ''p'', C, for some prime number ''p'', and natural number ''m''.
Every f ...
s. Denote the torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
of ''G'' as ''tG''. Then, ''G/tG'' is a torsion-free abelian group
In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only elemen ...
and thus it is free abelian. ''tG'' is a direct summand
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of ''G'', which means there exists a subgroup ''F'' of ''G'' s.t. , where . Then, ''F'' is also free abelian. Since ''tG'' is finitely generated and each element of ''tG'' has finite order, ''tG'' is finite. By the basis theorem for finite abelian group, ''tG'' can be written as direct sum of primary cyclic groups.
Invariant factor decomposition
We can also write any finitely generated abelian group ''G'' as a direct sum of the form
:
where ''k''1 divides
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 ...
''k''2, which divides ''k''3 and so on up to ''k''''u''. Again, the rank ''n'' and the ''invariant factor The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain.
If R is a PID and M a finitely generated R-module, then
:M\cong R^r\o ...
s'' ''k''1, ..., ''k''''u'' are uniquely determined by ''G'' (here with a unique order). The rank and the sequence of invariant factors determine the group up to isomorphism.
Equivalence
These statements are equivalent as a result of the Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
, which implies that if and only if ''j'' and ''k'' are coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
.
History
The history and credit for the fundamental theorem is complicated by the fact that it was proven when group theory was not well-established, and thus early forms, while essentially the modern result and proof, are often stated for a specific case. Briefly, an early form of the finite case was proven in , the finite case was proven in , and stated in group-theoretic terms in . The finitely ''presented'' case is solved by 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 ...
, and hence frequently credited to , though the finitely ''generated'' case is sometimes instead credited to ; details follow.
Group theorist László Fuchs László Fuchs (born June 24, 1924) is a Hungarian-born American mathematician, the Evelyn and John G. Phillips Distinguished Professor Emeritus in Mathematics at Tulane University. states:
The fundamental theorem for ''finite'' abelian groups was proven by Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
in , using a group-theoretic proof, though without stating it in group-theoretic terms; a modern presentation of Kronecker's proof is given in , 5.2.2 Kronecker's Theorem
176–177
This generalized an earlier result of Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
from ''Disquisitiones Arithmeticae
The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
'' (1801), which classified quadratic forms; Kronecker cited this result of Gauss's. The theorem was stated and proved in the language of groups by Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...
and Ludwig Stickelberger
Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomi ...
in 1878. Another group-theoretic formulation was given by Kronecker's student Eugen Netto
Eugen Otto Erwin Netto (30 June 1848 – 13 May 1919) was a German mathematician. He was born in Halle and died in Giessen.
Netto's theorem, on the dimension-preserving properties of continuous bijections, is named for Netto. Netto published ...
in 1882.[Wussing (2007), pp]
234–235
/ref>
The fundamental theorem for ''finitely presented'' abelian groups was proven by Henry John Stephen Smith
Prof Henry John Stephen Smith Royal Society of London, FRS FRSE Royal Astronomical Society, FRAS LLD (2 November 1826 – 9 February 1883) was an Irish mathematician and amateur astronomer remembered for his work in elementary divisors, quadrati ...
in , as integer matrices correspond to finite presentations of abelian groups (this generalizes to finitely presented modules over a principal ideal domain), and 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 ...
corresponds to classifying finitely presented abelian groups.
The fundamental theorem for ''finitely generated'' abelian groups was proven by Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
in , using a matrix proof (which generalizes to principal ideal domains). This was done in the context of computing the
homology
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 ...
of a complex, specifically the Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
and torsion coefficient
A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportional ...
s of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part.
Kronecker's proof was generalized to ''finitely generated'' abelian groups by Emmy Noether in .
Corollaries
Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
of finite 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
* H ...
and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
of ''G''. The rank of ''G'' is defined as the rank of the torsion-free part of ''G''; this is just the number ''n'' in the above formulas.
A corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
to the fundamental theorem is that every finitely generated torsion-free abelian group
In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only elemen ...
is free abelian. The finitely generated condition is essential here: is torsion-free but not free abelian.
Every 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 ...
and factor group
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the 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)
wh ...
s, form an abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
which is a Serre subcategory In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category.
Serre subca ...
of the category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of Ab is ...
.
Non-finitely generated abelian groups
Note that not every abelian group of finite rank is finitely generated; the rank 1 group is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of is another one.
See also
* The composition series in the Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...
is a non-abelian generalization.
Notes
References
* Reprinted (pp
367–409
i
''The Collected Mathematical Papers of Henry John Stephen Smith'', Vol. I
edited by J. W. L. Glaisher. Oxford: Clarendon Press (1894), ''xcv''+603 pp.
*
*
{{DEFAULTSORT:Finitely Generated Abelian Group
Abelian group theory
Algebraic structures