Prüfer group


In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, 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 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, 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.
There is a presentation
Here, the group operation in Z 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:
.
For each natural number n, consider the quotient group Z/pnZ and the embedding Z/pnZ → Z/pn+1Z induced by multiplication by p. The direct limit of this system is Z:
We can also write
where Qp denotes the additive group of p-adic numbers and Zp is the subgroup of p-adic integers.

Properties

The complete list of subgroups of the Prüfer p-group Z = Z/Z is:
with pn elements; it contains precisely those elements of Z 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. 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. As seen above, all proper subgroups of Z 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 number of copies of Q and numbers of copies of Z for every prime p. The numbers of copies of Q and Z that are used in this direct sum determine the divisible group up to isomorphism.
As an abelian group, Z is Artinian but not Noetherian. It can thus be used as a counterexample against the idea that every Artinian module is Noetherian.
The endomorphism ring of Z is isomorphic to the ring of p-adic integers Zp.
In the theory of locally compact topological groups the Prüfer p-group 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 group as a ring

As both the integers and the p-adic rationals are rings in addition to groups, the quotient ring is the Prüfer p-group with a ring structure, or the Prüfer p-ring.
Equivalently, could also be defined as the direct limit of the system of rings, where the homomorphisms are induced by multiplication by. This second definition allows a positional numeral system representation of the circle ring in base.
The product ring of and the integers yields the ring of p-adic rationals, while the product ring of and the p-adic integers yields the p-adic numbers.