# 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 different p-th roots. The group is named after Heinz Prüfer. It is a countable abelian group which helps taxonomize infinite abelian groups.

## Constructions of Z(p∞)

The Prüfer p-group may be identified with the subgroup of the circle group, U(1), consisting of all pn-th roots of unity as n ranges over all non-negative integers:

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

The group operation here is the multiplication of complex numbers.

Alternatively, the Prüfer p-group may be seen as the Sylow p-subgroup of the quotient group Q/Z, consisting of those elements whose order is a power of p:

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

(where Z[1/p] denotes the group of all rational numbers whose denominator is a power of p, using addition of rational numbers as group operation).

We can also write

$\mathbf{Z}(p^\infty)=\mathbf{Q}_p/\mathbf{Z}_p$

where Qp denotes the additive group of p-adic numbers and Zp is the subgroup of p-adic integers.

There is a presentation

$\mathbf{Z}(p^\infty) = \langle\, g_1, g_2, g_3, \ldots \mid g_1^p = 1, g_2^p = g_1, g_3^p = g_2, \dots\,\rangle.$

Here, the group operation in Z(p) is written as multiplication.

## Properties

The Prüfer p-groups for all primes p are the only infinite groups whose subgroups are totally ordered by inclusion:

$0 \subset \left({1 \over p}\mathbf{Z}\right)/\mathbf{Z} \subset \left({1 \over p^2}\mathbf{Z}\right)/\mathbf{Z} \subset \left({1 \over p^3}\mathbf{Z}\right)/\mathbf{Z} \subset \cdots \subset \mathbf{Z}(p^\infty)$

(Here $\left({1 \over p^n}\mathbf{Z}\right)/\mathbf{Z}$ is a cyclic subgroup of Z(p) with pn elements; it contains precisely those elements of Z(p) whose order divides pn and corresponds to the set of pn-th roots of unity.) This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup.

Given this list of subgroups, it is clear that the Prüfer p-groups are indecomposable (cannot be written as a direct sum of proper subgroups). More is true: the Prüfer p-groups are subdirectly irreducible. An abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic p-group or to a Prüfer group.

The Prüfer p-group is the unique infinite p-group which is locally cyclic (every finite set of elements generates a cyclic group). As seen above, all proper subgroups of Z(p) are finite. The Prüfer p-groups are the only infinite abelian groups with this property.[1]

The Prüfer p-groups are divisible. They play an important role in the classification of divisible groups; along with the rational numbers they are the simplest divisible groups. More precisely: an abelian group is divisible if and only if it is the direct sum of a (possibly infinite) number of copies of Q and (possibly infinite) numbers of copies of Z(p) for every prime p. The numbers of copies of Q and Z(p) that are used in this direct sum determine the divisible group up to isomorphism.[2]

As an abelian group (that is, as a Z-module), Z(p) is Artinian but not Noetherian.[3] It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian ring is Noetherian).

The endomorphism ring of Z(p) is isomorphic to the ring of p-adic integers Zp.[4]

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.[5]