Prüfer group

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:

The group operation here is the multiplication of complex numbers.

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:

(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

where Q_p denotes the additive group of p-adic numbers and Z_p is the subgroup of p-adic integers.

There is a presentation

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

Properties

The complete list of subgroups of the Prüfer p-group Z(p^∞) is:

(Here ( 1 p n Z ) / 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.) 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 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.

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.