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| [[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''>]]
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| 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.
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| 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:
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| :<math>\mathbf{Z}(p^\infty)=\{\exp(2\pi i m/p^n) \mid m\in \mathbf{Z}^+,\,n\in \mathbf{Z}^+\}.\;</math>
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| 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'':
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| :<math>\mathbf{Z}(p^\infty) = \mathbf{Z}[1/p]/\mathbf{Z}</math>
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| or equivalently
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| <math>\mathbf{Z}(p^\infty)=\mathbf{Q}_p/\mathbf{Z}_p.</math>
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| There is a [[Group presentation|presentation]]
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| :<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>
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| 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).
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| The Prüfer ''p''-group is [[divisible group|divisible]].
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| 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.
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| 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>
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| 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]].
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| :<math>0 \subset \mathbf{Z}/p \subset \mathbf{Z}/p^2 \subset \mathbf{Z}/p^3 \subset \cdots \subset \mathbf{Z}(p^\infty)</math>
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| This sequence of inclusions expresses the Prüfer ''p''-group as the [[direct limit]] of its finite subgroups.
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| 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).
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| ==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.
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| * [[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.
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| ==Notes==
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| <references/>
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| ==References==
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| * {{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}}}}
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| * {{cite book|author=Pierre Antoine Grillet|title=Abstract algebra|year=2007|publisher=Springer|isbn=978-0-387-71567-4}}
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| * {{planetmath reference|id=7500|title=Quasicyclic group}}
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| * {{springer|id=Q/q076440|author=N.N. Vil'yams|title=Quasi-cyclic group}}
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| {{DEFAULTSORT:Prufer Group}}
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| [[Category:Abelian group theory]]
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| [[Category:Infinite group theory]]
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| [[Category:P-groups]]
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Oscar is how he's called and he totally loves this name. One of the issues he loves most is ice skating but he is having difficulties to discover time for it. For a while she's been in South Dakota. Hiring has been my profession for some time but I've already applied for an additional 1.
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