Editing Finitely generated abelian group

{{short description|Commutative group where every element is the sum of elements from one finite subset}}
In [[abstract algebra]], an [[abelian group]] <math>(G,+)</math> is called finitely generated if there exist finitely many elements <math>x_1,\dots,x_s</math> in <math>G</math> such that every <math>x</math> in <math>G</math> can be written in the form <math>x = n_1x_1 + n_2x_2 + \cdots + n_sx_s</math> for some [[integer]]s <math>n_1,\dots, n_s</math>. In this case, we say that the set <math>\{x_1,\dots, x_s\}</math> is a ''[[generating set of a group|generating set]]'' of <math>G</math> or that <math>x_1,\dots, x_s</math> ''generate'' <math>G</math>. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups.

Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.

==Examples==
* The [[integers]], <math>\left(\mathbb{Z},+\right)</math>, are a finitely generated abelian group.
* The [[modular arithmetic|integers modulo <math>n</math>]], <math>\left(\mathbb{Z}/n\mathbb{Z},+\right)</math>, are a finite (hence finitely generated) abelian group.
* Any [[direct sum]] of finitely many finitely generated abelian groups is again a finitely generated abelian group.
* Every [[Lattice (group)|lattice]] forms a finitely generated [[free abelian group]].

There are no other examples (up to isomorphism). In particular, the group <math>\left(\mathbb{Q},+\right)</math> of [[rational number]]s is not finitely generated:<ref name="Silverman-Tate-1992">Silverman & Tate (1992), [{{Google books|plainurl=y|id=mAJei2-JcE4C|page=102|text=not finitely generated}} p. 102]</ref> if <math>x_1,\ldots,x_n</math> are rational numbers, pick a [[natural number]] <math>k</math> [[coprime]] to all the denominators; then <math>1/k</math> cannot be generated by <math>x_1,\ldots,x_n</math>. The group <math>\left(\mathbb{Q}^*,\cdot\right)</math> of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition <math> \left(\mathbb{R},+\right)</math> and non-zero real numbers under multiplication <math>\left(\mathbb{R}^*,\cdot\right)</math> are also not finitely generated.<ref name="Silverman-Tate-1992" /><ref>de la Harpe (2000), [{{Google books|plainurl=y|id=60fTzwfqeQIC|page=46|text=The multiplicative group Q}} p. 46]</ref>

==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|fundamental theorem of ''finite'' abelian groups]]. The theorem, in both forms, in turn generalizes to the [[structure theorem for finitely generated modules over a principal ideal domain]], 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]] of [[primary cyclic group]]s and infinite [[cyclic group]]s. A primary cyclic group is one whose [[order of a group|order]] is a power of a [[prime number|prime]]. That is, every finitely generated abelian group is isomorphic to a group of the form
:<math>\mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z},</math>
where ''n'' ≥ 0 is the ''[[Rank of an abelian group|rank]]'', and the numbers ''q''<sub>1</sub>, ..., ''q''<sub>''t''</sub> are powers of (not necessarily distinct) prime numbers. In particular, ''G'' is finite if and only if ''n'' = 0.  The values of ''n'', ''q''<sub>1</sub>, ..., ''q''<sub>''t''</sub> are ([[up to]] 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]]: every finite abelian group is a [[direct sum]] of [[primary cyclic group]]s. Denote the [[torsion subgroup]] of ''G'' as ''tG''. Then, ''G/tG'' is a [[torsion-free abelian group]] and thus it is free abelian. ''tG'' is a [[direct summand]] of ''G'', which means there exists a subgroup ''F'' of ''G'' s.t. <math>G=tG\oplus F</math>, where <math>F\cong G/tG</math>. 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
:<math>\mathbb{Z}^n \oplus \mathbb{Z}/{k_1}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/{k_u}\mathbb{Z},</math>
where ''k''<sub>1</sub> [[divisor|divides]] ''k''<sub>2</sub>, which divides ''k''<sub>3</sub> and so on up to ''k''<sub>''u''</sub>.  Again, the rank ''n'' and the ''[[invariant factor]]s'' ''k''<sub>1</sub>, ..., ''k''<sub>''u''</sub> 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]], which implies that <math>\mathbb{Z}_{jk}\cong \mathbb{Z}_{j} \oplus \mathbb{Z}_{k}</math> if and only if ''j'' and ''k'' are [[coprime]].

===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 by Gauss in 1801, the finite case was proven by  Kronecker in 1870, and stated in group-theoretic terms by Frobenius and Stickelberger in 1878.{{CN|date=March 2023}} The [[Finitely presented group|finitely ''presented'']] case is solved by [[Smith normal form]], and hence frequently credited to {{harv|Smith|1861}},<ref name="fuchs"/> though the finitely ''generated'' case is sometimes instead credited to Poincaré in 1900;{{CN|date=March 2023}} details follow.

Group theorist [[László Fuchs]] states:<ref name=fuchs>{{cite book
|title=Abelian Groups
|first=László
|last=Fuchs
|author-link=László Fuchs
|isbn=978-3-319-19422-6
|year=2015
|orig-year=Originally published 1958
|page=[https://books.google.com/books?id=2KMvCwAAQBAJ&pg=PA85 85]
|publisher=Springer
}}</ref>
{{quote|As far as the fundamental theorem on finite abelian groups is concerned, it is not clear how far back in time one needs to go to trace its origin. ... it took a long time to formulate and prove the fundamental theorem in its present form ... }}

The fundamental theorem for ''finite'' abelian groups was proven by [[Leopold Kronecker]] in 1870,{{CN|date=March 2023}} using a group-theoretic proof,<ref name=stillwell175>{{cite book 
|title=Classical Topology and Combinatorial Group Theory
|first=John
|last=Stillwell
|author-link=John Stillwell
|year=2012
|section=5.2 The Structure Theorem for Finitely Generated Abelian Groups 
|page=[https://books.google.com/books?id=WtcRBwAAQBAJ&pg=PA175 175]
}}</ref> though without stating it in group-theoretic terms;<ref name=wussing67>{{cite book
|first=Hans
|last=Wussing
|author-link=Hans Wussing
|title=Die Genesis des abstrakten Gruppenbegriffes. Ein Beitrag zur Entstehungsgeschichte der abstrakten Gruppentheorie.
|orig-year=1969
|trans-title=The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory.
|year=2007
|page=[https://books.google.com/books?id=Xp3JymnfAq4C&pg=PA67 67]
}}</ref> a modern presentation of Kronecker's proof is given in {{harv|Stillwell|2012}}, 5.2.2 Kronecker's Theorem, [https://books.google.com/books?id=WtcRBwAAQBAJ&pg=PA176 176–177]. This generalized an earlier result of [[Carl Friedrich Gauss]] from ''[[Disquisitiones Arithmeticae]]'' (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]] and [[Ludwig Stickelberger]] in 1878.<ref>G. Frobenius, L. Stickelberger, Ueber Gruppen von vertauschbaren Elementen, J. reine u. angew. Math., 86 (1878), 217-262.</ref><ref>Wussing (2007), pp. [https://books.google.com/books?id=Xp3JymnfAq4C&pg=PA234 234–235]</ref> Another group-theoretic formulation was given by Kronecker's student [[Eugen Netto]] in 1882.<ref>''Substitutionentheorie und ihre Anwendung auf die Algebra'',
Eugen Netto, 1882</ref><ref>Wussing (2007), pp. [https://books.google.com/books?id=Xp3JymnfAq4C&pg=PA234 234–235]</ref>

The fundamental theorem for [[Finitely presented group|''finitely presented'']] abelian groups was proven by [[Henry John Stephen Smith]] in {{harv|Smith|1861}},<ref name="fuchs"/> 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]] corresponds to classifying finitely presented abelian groups.

The fundamental theorem for ''finitely generated'' abelian groups was proven by [[Henri Poincaré]] in 1900, using a matrix proof (which generalizes to principal ideal domains).{{CN|date=March 2023}} This was done in the context of computing the
[[Homology (mathematics)|homology]] of a complex, specifically the [[Betti number]] and [[Torsion coefficient (topology)|torsion coefficient]]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.<ref name=stillwell175 />

Kronecker's proof was generalized to ''finitely generated'' abelian groups by Emmy Noether in 1926.<ref name=stillwell175 />

==Corollaries==
Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a [[free abelian group]] of finite [[rank of an abelian group|rank]] and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the [[torsion subgroup]] 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]] to the fundamental theorem is that every finitely generated [[torsion-free abelian group]] is free abelian. The finitely generated condition is essential here: <math>\mathbb{Q}</math> is torsion-free but not free abelian.

Every [[subgroup]] and [[factor group]] of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the [[group homomorphism]]s, form an [[abelian category]] which is a [[Subcategory#Types_of_subcategories|Serre subcategory]] of the [[category of abelian groups]].

==Non-finitely generated abelian groups==
Note that not every abelian group of finite rank is finitely generated; the rank 1 group <math>\mathbb{Q}</math> is one counterexample, and the rank-0 group given by a direct sum of [[infinite set|countably infinitely many]] copies of <math>\mathbb{Z}_{2}</math> is another one.

==See also==
* The composition series in the [[Jordan–Hölder theorem]] is a non-abelian generalization.

==Notes==
{{Reflist}}

==References==
{{refbegin}}
* {{cite journal |last=Smith |first=Henry J. Stephen |author-link=Henry John Stephen Smith |year=1861 |title=On systems of linear indeterminate equations and congruences |journal=[[Philosophical Transactions of the Royal Society of London|Phil. Trans. R. Soc. Lond.]] |volume=151 |issue=1 |pages=293–326 |jstor=108738 |doi=10.1098/rstl.1861.0016 |s2cid=110730515 }} Reprinted (pp. [https://archive.org/stream/collectedmathema01smituoft#page/366/mode/2up 367–409]) in [https://archive.org/details/collectedmathema01smituoft ''The Collected Mathematical Papers of Henry John Stephen Smith'', Vol. I], edited by [[James Whitbread Lee Glaisher|J. W. L. Glaisher]]. Oxford: Clarendon Press (1894), ''xcv''+603 pp.
* {{cite book |last1=Silverman |first1=Joseph H. |last2=Tate |first2=John Torrence |title=Rational points on elliptic curves |series=[[Undergraduate Texts in Mathematics]] |year=1992 |publisher=Springer |isbn=978-0-387-97825-3 }}
* {{cite book |last1=de la Harpe |first1=Pierre |title=Topics in geometric group theory |series=Chicago lectures in mathematics |year=2000 |publisher=University of Chicago Press |isbn=978-0-226-31721-2 }}
{{refend}}


{{DEFAULTSORT:Finitely Generated Abelian Group}}
[[Category:Abelian group theory]]
[[Category:Algebraic structures]]