formulasearchengine - User contributions [en]

Rabin cryptosystem

2014-11-03T20:48:53Z

129.137.166.167: /* Computing square roots */

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Feige–Fiat–Shamir identification scheme

2014-09-26T19:44:00Z

129.137.166.167: /* Setup */

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Chord (aircraft)

2014-02-03T20:01:25Z

129.137.170.55: /* Mean aerodynamic chord */

In [[abstract algebra]], an [[abelian group]] (''G'',+) is called '''finitely generated''' if there exist finitely many elements ''x''1,...,''x''''s'' in ''G'' such that every ''x'' in ''G'' can be written in the form
:''x'' = ''n''1''x''1 + ''n''2''x''2 + ... + ''n''''s''''x''''s''
with [[integer]]s ''n''1,...,''n''''s''. In this case, we say that the set {''x''1,...,''x''''s''} is a ''[[generating set of a group|generating set]]'' of ''G'' or that ''x''1,...,''x''''s'' ''generate'' ''G''.

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.

== Examples ==
* the [[integers]] <math>\left(\mathbb{Z},+\right)</math> are a finitely generated abelian group
* the [[modular arithmetic|integers modulo <math>n</math>]], <math>\mathbb{Z}_n</math> are a finitely generated abelian group
* any [[Direct sum of groups|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 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>

== Classification ==
The '''fundamental theorem of finitely generated abelian groups'''
(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):

===Primary decomposition===
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
:<math>\mathbb{Z}^n \oplus \mathbb{Z}_{q_1} \oplus \cdots \oplus \mathbb{Z}_{q_t},</math>
where the ''[[Rank of an abelian group|rank]]'' ''n'' ≥ 0, 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]] rearranging the indices) uniquely determined by ''G''.

===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} \oplus \cdots \oplus \mathbb{Z}_{k_u},</math>
where ''k''1 [[divisor|divides]] ''k''2, which divides ''k''3 and so on up to ''k''''u''. Again, the rank ''n'' and the ''[[invariant factor]]s'' ''k''1,...,''k''''u'' are uniquely determined by ''G'' (here with a unique order).

===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''.

==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|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 [[Jordan–Hölder theorem]] is a non-abelian generalization

== Notes ==
{{Reflist}}

== References ==
* {{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= }}
* {{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= }}

{{fundamental theorems}}

{{DEFAULTSORT:Finitely-Generated Abelian Group}}
[[Category:Abelian group theory]]
[[Category:Fundamental theorems|*Finitely-generated abelian group]]
[[Category:Algebraic structures]]

Osmotic concentration

2013-09-02T14:49:37Z

129.137.155.17:

'''Cassie's law''' describes the effective [[contact angle]] [[theta|θ]]c for a liquid on a [[composite material|composite surface]].<ref>{{cite journal|doi=10.1039/TF9444000546|title=Wettability of porous surfaces|year=1944|last1=Cassie|first1=A. B. D.|last2=Baxter|first2=S.|journal=Transactions of the Faraday Society|volume=40|pages=546}}</ref> The law explains how simply roughing up a surface increases the apparent surface angle. The law is stated as:

:<math> \cos \theta_c =\mathit{f}_1 \cos \theta_1 + \mathit{f}_2 \cos \theta_2 \,</math>

where by [[theta|θ]]1 is the contact angle for component 1 with [[areal]] fraction f1 and [[theta|θ]]2 is the contact angle for component 2 with areal fraction f2 present in the composite material. This equation takes on special meaning when in a 2-component system one component is [[air]] with a [[contact angle]] of 180[[Degree (angle)|°]]. With [[cosine]](180) = –1, the equation reduces to:

:<math> \cos \theta_c =\mathit{f}_1 ( \cos \theta_1 + 1) -1 \,</math>

which implies that with a small f1 and a large [[theta|θ]]1 it is possible to create surfaces with a very large contact angle. Cassie's research pointed out that the water repelling quality of [[duck]]s is due to the very nature of the composite formed between air and feather and not by other causes such as the presence of exceptional proofing agents like oils. [[Water strider]]s also exploit this phenomenon. Artificial [[superhydrophobic material]]s such as [[nanopin film]] exist in the laboratory that also make use of this law.

==See also==
*[[Goniometry]]

==References==
{{reflist}}

{{DEFAULTSORT:Cassie's Law}}
[[Category:Condensed matter physics]]

Vortex lattice method

2012-07-17T18:04:18Z

129.137.198.49:

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Babuška–Lax–Milgram theorem

2012-04-02T16:44:37Z

129.137.179.174: /* Statement of the theorem */

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