# Prüfer group

The Prüfer 2-group. <gn: gn+12 = gn, 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:

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

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.[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.

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

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.[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).

2. ^ Subgroups of an abelian group are abelian, and coincide with submodules as a $\mathbf{Z}$-module.