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- Prüfer_group abstract "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:or equivalentlyThere is a presentationThe 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.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. It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian ring is Noetherian).".
- Prüfer_group thumbnail Prüfer.png?width=300.
- Prüfer_group wikiPageID "4885031".
- Prüfer_group wikiPageRevisionID "597640978".
- Prüfer_group author "N.N. Vil'yams".
- Prüfer_group id "7500".
- Prüfer_group id "Q/q076440".
- Prüfer_group title "Quasi-cyclic group".
- Prüfer_group title "Quasicyclic group".
- Prüfer_group subject Category:Abelian_group_theory.
- Prüfer_group subject Category:Infinite_group_theory.
- Prüfer_group subject Category:P-groups.
- Prüfer_group comment "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.".
- Prüfer_group label "Groupe de Prüfer".
- Prüfer_group label "Grupa Prüfera".
- Prüfer_group label "Grupo de Prüfer".
- Prüfer_group label "Grupo de Prüfer".
- Prüfer_group label "Gruppo di Prüfer".
- Prüfer_group label "Prüfer group".
- Prüfer_group label "Prüfer-groep".
- Prüfer_group sameAs Pr%C3%BCfer_group.
- Prüfer_group sameAs Prüferova_grupa.
- Prüfer_group sameAs Grupo_de_Prüfer.
- Prüfer_group sameAs Groupe_de_Prüfer.
- Prüfer_group sameAs Gruppo_di_Prüfer.
- Prüfer_group sameAs Prüfer-groep.
- Prüfer_group sameAs Grupa_Prüfera.
- Prüfer_group sameAs Grupo_de_Prüfer.
- Prüfer_group sameAs Q2281199.
- Prüfer_group sameAs Q2281199.
- Prüfer_group wasDerivedFrom Prüfer_group?oldid=597640978.
- Prüfer_group depiction Prüfer.png.
