Nullcline

The Prüfer 2-group. <g_n: g_n+1² = g_n, g₁² = e>

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:

$\mathbf {Z} (p^{\infty })=\{\exp(2\pi im/p^{n})\mid m\in \mathbf {Z} ^{+},\,n\in \mathbf {Z} ^{+}\}.\;$

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:

$\mathbf {Z} (p^{\infty })=\mathbf {Z} [1/p]/\mathbf {Z}$

or equivalently $\mathbf {Z} (p^{\infty })=\mathbf {Q} _{p}/\mathbf {Z} _{p}.$

There is a presentation

$\mathbf {Z} (p^{\infty })=\langle \,x_{1},x_{2},x_{3},\ldots \mid x_{1}^{p}=1,x_{2}^{p}=x_{1},x_{3}^{p}=x_{2},\dots \,\rangle .$

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.

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.

$0\subset \mathbf {Z} /p\subset \mathbf {Z} /p^{2}\subset \mathbf {Z} /p^{3}\subset \cdots \subset \mathbf {Z} (p^{\infty })$

This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups.

As a $\mathbf {Z}$-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).

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/2^{b. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1.}

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Template:Planetmath reference
• Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
  
  my web-site http://himerka.com/