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.
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 divisible.
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 an Abelian group (that is, as a -module), 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).
- 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.
- 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 so are submodules of -modules.
- See also Jacobson (2009), p. 102, ex. 2.
- Jacobson, Nathan (2009). Basic algebra 2 (2nd ed.). Dover. ISBN 978-0-486-47187-7
- Pierre Antoine Grillet (2007). Abstract algebra. Springer. ISBN 978-0-387-71567-4.
- Quasicyclic group at PlanetMath.org.
- N.N. Vil'yams (2001), "Quasi-cyclic group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4