Nullcline: Difference between revisions
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The | [[File:Prüfer.png|thumb|300px|The Prüfer 2-group. <''g''<sub>''n''</sub>: ''g''<sub>''n''+1</sub><sup>2</sup> = ''g''<sub>''n''</sub>, ''g''<sub>1</sub><sup>2</sup> = ''e''>]] | ||
In [[mathematics]], specifically in [[group theory]], the '''Prüfer ''p''-group''' or the '''''p''-quasicyclic group''' or '''''p''<sup>∞</sup>'''-group, '''Z'''(''p''<sup>∞</sup>), for a [[prime number]] ''p'' is the unique [[p-group|''p''-group]] in which every element has ''p'' ''p''th roots. The group is named after [[Heinz Prüfer]]. It is a [[countable set|countable]] [[abelian group]] which helps [[Taxonomy (general)|taxonomize]] infinite abelian groups. | |||
The Prüfer ''p''-group may be [[group representation|represented]] as a subgroup of the [[circle group]], U(1), as the set of ''p''<sup>''n''</sup>th [[root of unity|roots of unity]] as ''n'' ranges over all non-negative integers: | |||
:<math>\mathbf{Z}(p^\infty)=\{\exp(2\pi i m/p^n) \mid m\in \mathbf{Z}^+,\,n\in \mathbf{Z}^+\}.\;</math> | |||
Alternatively, the Prüfer ''p''-group may be seen as the [[Sylow subgroup|Sylow p-subgroup]] of '''Q''/''Z''', consisting of those elements whose order is a power of ''p'': | |||
:<math>\mathbf{Z}(p^\infty) = \mathbf{Z}[1/p]/\mathbf{Z}</math> | |||
or equivalently | |||
<math>\mathbf{Z}(p^\infty)=\mathbf{Q}_p/\mathbf{Z}_p.</math> | |||
There is a [[Group presentation|presentation]] | |||
:<math>\mathbf{Z}(p^\infty) = \langle\, x_1, x_2, x_3, \ldots \mid x_1^p = 1, x_2^p = x_1, x_3^p = x_2, \dots\,\rangle.</math> | |||
The Prüfer ''p''-group is the unique infinite [[p-group|''p''-group]] which is [[locally cyclic group|locally cyclic]] (every finite set of elements generates a cyclic group). | |||
The Prüfer ''p''-group is [[divisible group|divisible]]. | |||
In the language of [[universal algebra]], an abelian group is [[subdirectly irreducible algebra|subdirectly irreducible]] if and only if it is isomorphic to a finite cyclic ''p''-group or isomorphic to a Prüfer group. | |||
In the theory of [[locally compact topological group]]s the Prüfer ''p''-group (endowed with the discrete topology) is the [[Pontryagin dual]] of the compact group of [[p-adic integer]]s, and the group of ''p''-adic integers is the Pontryagin dual of the Prüfer ''p''-group.<ref>D. L. Armacost and W. L. Armacost, "[http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102968274 On ''p''-thetic groups]", ''Pacific J. Math.'', '''41''', no. 2 (1972), 295–301</ref> | |||
The Prüfer ''p''-groups for all primes ''p'' are the only infinite groups whose subgroups are [[totally ordered]] by inclusion. As there is no [[maximal subgroup]] of a Prüfer ''p''-group, it is its own [[Frattini subgroup]]. | |||
:<math>0 \subset \mathbf{Z}/p \subset \mathbf{Z}/p^2 \subset \mathbf{Z}/p^3 \subset \cdots \subset \mathbf{Z}(p^\infty)</math> | |||
This sequence of inclusions expresses the Prüfer ''p''-group as the [[direct limit]] of its finite subgroups. | |||
As a <math>\mathbf{Z}</math>-module, the Prüfer ''p''-group is [[Artinian module|Artinian]], but not [[Noetherian module|Noetherian]], and likewise as a group, it is [[Artinian group|Artinian]] but not [[Noetherian group|Noetherian]].<ref>Subgroups of an abelian group are abelian, and coincide with submodules as a <math>\mathbf{Z}</math>-module.</ref><ref>See also Jacobson (2009), p. 102, ex. 2.</ref> It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every [[Artinian ring|Artinian ''ring'']] is Noetherian). | |||
==See also== | |||
* [[p-adic integers|''p''-adic integers]], which can be defined as the [[inverse limit]] of the finite subgroups of the Prüfer ''p''-group. | |||
* [[Dyadic rational]], rational numbers of the form ''a''/2<sup>''b''</sub>. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1. | |||
==Notes== | |||
<references/> | |||
==References== | |||
* {{Cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 2 | series= | publisher=Dover| isbn = 978-0-486-47187-7| postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}} | |||
* {{cite book|author=Pierre Antoine Grillet|title=Abstract algebra|year=2007|publisher=Springer|isbn=978-0-387-71567-4}} | |||
* {{planetmath reference|id=7500|title=Quasicyclic group}} | |||
* {{springer|id=Q/q076440|author=N.N. Vil'yams|title=Quasi-cyclic group}} | |||
{{DEFAULTSORT:Prufer Group}} | |||
[[Category:Abelian group theory]] | |||
[[Category:Infinite group theory]] | |||
[[Category:P-groups]] |
Revision as of 16:00, 20 November 2013
In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer. It is a countable abelian group which helps taxonomize infinite abelian groups.
The Prüfer p-group may be represented as a subgroup of the circle group, U(1), as the set of pnth roots of unity as n ranges over all non-negative integers:
Alternatively, the Prüfer p-group may be seen as the Sylow p-subgroup of Q/Z, consisting of those elements whose order is a power of p:
There is a presentation
The Prüfer p-group is the unique infinite p-group which is locally cyclic (every finite set of elements generates a cyclic group).
The Prüfer p-group is divisible.
In the language of universal algebra, an abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic p-group or isomorphic to a Prüfer group.
In the theory of locally compact topological groups the Prüfer p-group (endowed with the discrete topology) is the Pontryagin dual of the compact group of p-adic integers, and the group of p-adic integers is the Pontryagin dual of the Prüfer p-group.[1]
The Prüfer p-groups for all primes p are the only infinite groups whose subgroups are totally ordered by inclusion. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup.
This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups.
As a -module, the Prüfer p-group is Artinian, but not Noetherian, and likewise as a group, it is Artinian but not Noetherian.[2][3] It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian ring is Noetherian).
See also
- p-adic integers, which can be defined as the inverse limit of the finite subgroups of the Prüfer p-group.
- Dyadic rational, rational numbers of the form a/2b. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1.
Notes
- ↑ D. L. Armacost and W. L. Armacost, "On p-thetic groups", Pacific J. Math., 41, no. 2 (1972), 295–301
- ↑ Subgroups of an abelian group are abelian, and coincide with submodules as a -module.
- ↑ See also Jacobson (2009), p. 102, ex. 2.
References
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