# Prüfer group

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 Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups.
The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.

## Constructions of Z(p$^∞$)

The Prüfer p-group may be identified with the subgroup of the circle group, U(1), consisting of all p$^n$-th roots of unity as n ranges over all non-negative integers:
    $\mathbf {Z} (p^{\infty })=\{\exp(2\pi im/p^{n})\mid 0\leq m<p^{n},\,n\in \mathbf {Z} ^{+}\}=\{z\in \mathbf {C} \mid z^{(p^{n})}=1{\text{ for some }}n\in \mathbf {Z} ^{+}\}.\;$
The group operation here is the multiplication of complex numbers.
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.
Alternatively and equivalently, the Prüfer p-group may be defined 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).
For each natural number n, consider the quotient group Z/p$^n$Z and the embedding Z/p$^n$Z → Z/p$^{n+1}$Z induced by multiplication by p. The direct limit of this system is Z(p$^∞$):
    $\mathbf {Z} (p^{\infty })=\varinjlim \mathbf {Z} /p^{n}\mathbf {Z} .$
If we perform the direct limit in the category of topological groups, then we need to impose a topology on each of the $\mathbf {Z} /p^{n}\mathbf {Z}$, and take the final topology on $\mathbf {Z} (p^{\infty })$. If we wish for $\mathbf {Z} (p^{\infty })$ to be Hausdorff, we must impose the discrete topology on each of the $\mathbf {Z} /p^{n}\mathbf {Z}$, resulting in $\mathbf {Z} (p^{\infty })$ to have the discrete topology.
We can also write
    $\mathbf {Z} (p^{\infty })=\mathbf {Q} _{p}/\mathbf {Z} _{p}$
where Q$_p$ denotes the additive group of p-adic numbers and Z$_p$ is the subgroup of p-adic integers.

## Properties

The complete list of subgroups of the Prüfer p-group Z(p$^∞$) = Z[1/p]/Z is:
    $0\subsetneq \left({1 \over p}\mathbf {Z} \right)/\mathbf {Z} \subsetneq \left({1 \over p^{2}}\mathbf {Z} \right)/\mathbf {Z} \subsetneq \left({1 \over p^{3}}\mathbf {Z} \right)/\mathbf {Z} \subsetneq \cdots \subsetneq \mathbf {Z} (p^{\infty })$
Here, each $\left({1 \over p^{n}}\mathbf {Z} \right)/\mathbf {Z}$ is a cyclic subgroup of Z(p$^∞$) with p$^n$ elements; it contains precisely those elements of Z(p$^∞$) whose order divides p$^n$ and corresponds to the set of p$^n$-th roots of unity.
The Prüfer p-groups are the only infinite groups whose subgroups are totally ordered by inclusion. 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 that 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.
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 (cardinal) numbers of copies of Q and Z(p$^∞$) that are used in this direct sum determine the divisible group up to isomorphism.
As an abelian group (that is, as a Z-module), Z(p$^∞$) 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).
The endomorphism ring of Z(p$^∞$) is isomorphic to the ring of p-adic integers Z$_p$.
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.