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| In [[abstract algebra]], an [[abelian group]] (''G'',+) is called '''finitely generated''' if there exist finitely many elements ''x''<sub>1</sub>,...,''x''<sub>''s''</sub> in ''G'' such that every ''x'' in ''G'' can be written in the form | | In the past protection camera [http://www.badideafairy.com/biotronesis/index.php?title=Outrageous_Network_Camera_Dvr_Tips arrangements] are already monitored insurance agencies a separate monitor located about the premises where the cameras and DVR are pinpointed. CCTV for home within the UK can be found online quite easily with a lot of leading websites that sell home security systems. Cctv dvr pc cms software free download For males and females that know little reely about what a CCTV digicam is, or just what it does.<br><br> |
| :''x'' = ''n''<sub>1</sub>''x''<sub>1</sub> + ''n''<sub>2</sub>''x''<sub>2</sub> + ... + ''n''<sub>''s''</sub>''x''<sub>''s''</sub> | |
| with [[integer]]s ''n''<sub>1</sub>,...,''n''<sub>''s''</sub>. In this case, we say that the set {''x''<sub>1</sub>,...,''x''<sub>''s''</sub>} is a ''[[generating set of a group|generating set]]'' of ''G'' or that ''x''<sub>1</sub>,...,''x''<sub>''s''</sub> ''generate'' ''G''.
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| Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.
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| == Examples ==
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| * the [[integers]] <math>\left(\mathbb{Z},+\right)</math> are a finitely generated abelian group
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| * the [[modular arithmetic|integers modulo <math>n</math>]], <math>\mathbb{Z}_n</math> are a finitely generated abelian group
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| * any [[Direct sum of groups|direct sum]] of finitely many finitely generated abelian groups is again a finitely generated abelian group
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| * every [[Lattice (group)|lattice]] forms a finitely-generated [[free abelian group]]
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| 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 quote|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.<ref name="Silverman-Tate-1992" /><ref>La Harpe (2000), {{Google books quote|id=60fTzwfqeQIC|page=46|text=The multiplicative group Q|p. 46}}</ref>
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| == Classification ==
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| The '''fundamental theorem of finitely generated abelian groups'''
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| (which is a special case of the [[structure theorem for finitely generated modules over a principal ideal domain]]) can be stated two ways (analogously with [[principal ideal domain]]s):
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| ===Primary decomposition===
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| The primary decomposition formulation states that every finitely generated abelian group ''G'' is isomorphic to a [[Direct sum of groups|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
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| :<math>\mathbb{Z}^n \oplus \mathbb{Z}_{q_1} \oplus \cdots \oplus \mathbb{Z}_{q_t},</math>
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| where the ''[[Rank of an abelian group|rank]]'' ''n'' ≥ 0, 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''.
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| ===Invariant factor decomposition=== | |
| We can also write any finitely generated abelian group ''G'' as a direct sum of the form
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| :<math>\mathbb{Z}^n \oplus \mathbb{Z}_{k_1} \oplus \cdots \oplus \mathbb{Z}_{k_u},</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).
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| ===Equivalence=== | |
| These statements are equivalent because of the [[Chinese remainder theorem]], which here states that <math>\mathbb{Z}_{m}\simeq \mathbb{Z}_{j} \oplus \mathbb{Z}_{k}</math> if and only if ''j'' and ''k'' are [[coprime]] and ''m'' = ''jk''.
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| ==Corollaries==
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| 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.
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| 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.
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| 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|Serre subcategory]] of the [[category of abelian groups]].
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| ==Non-finitely generated abelian groups==
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| 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.
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| ==See also==
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| * The [[Jordan–Hölder theorem]] is a non-abelian generalization
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| * {{cite book |last1=Silverman |first1=Joseph H. |authorlink1= |last2=Tate |first2=John Torrence |authorlink2= |title=Rational points on elliptic curves |url= |edition= |series=Undergraduate texts in mathematics |volume= |year=1992 |publisher=Springer |location= |isbn=978-0-387-97825-3 |id= }}
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| * {{cite book |last1=La Harpe |first1=Pierre de |authorlink1= |last2= |first2= |authorlink2= |title=Topics in geometric group theory |url= |edition= |series=Chicago lectures in mathematics |volume= |year=2000 |publisher=University of Chicago Press |location= |isbn=978-0-226-31721-2 |id= }}
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| {{fundamental theorems}}
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| {{DEFAULTSORT:Finitely-Generated Abelian Group}}
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| [[Category:Abelian group theory]]
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| [[Category:Fundamental theorems|*Finitely-generated abelian group]]
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| [[Category:Algebraic structures]]
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