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:
:$\mathbf(p^\infty)=\ = \.\;$
The group operation here is the multiplication of complex numbers.

There is a presentation
:$\mathbf(p^\infty) = \langle\, g_1, g_2, g_3, \ldots \mid g_1^p = 1, g_2^p = g_1, g_3^p = g_2, \dots\,\rangle.$
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(p^\infty) = \mathbf /p\mathbf
(where Z /''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''+1Z induced by multiplication by ''p''. The direct limit of this system is Z(''p''^∞):
:$\mathbf(p^\infty) = \varinjlim \mathbf/p^n \mathbf .$

If we perform the direct limit in the category of topological groups, then we need to impose a topology on each of the $\mathbf/p^n \mathbf$, and take the final topology on $\mathbf(p^\infty)$. If we wish for $\mathbf(p^\infty)$ to be Hausdorff, we must impose the discrete topology on each of the $\mathbf/p^n \mathbf$, resulting in $\mathbf(p^\infty)$ to have the discrete topology.

We can also write
:$\mathbf(p^\infty)=\mathbf_p/\mathbf_p$
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 /''p''Z is:

:$0 \subsetneq \left(\mathbf\right)/\mathbf \subsetneq \left(\mathbf\right)/\mathbf \subsetneq \left(\mathbf\right)/\mathbf \subsetneq \cdots \subsetneq \mathbf(p^\infty)$
(Here $\left(\mathbf\right)/\mathbf$ 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 ''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 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.D. L. Armacost and W. L. Armacost
On ''p''-thetic groups
, ''Pacific J. Math.'', 41, no. 2 (1972), 295–301

See also

* ''p''-adic integers, which can be defined as the inverse limit of the finite subgroups of the Prüfer ''p''-group.
* Dyadic rational, rational numbers of the form ''a''/2^''b''. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1.
*Cyclic group (finite analogue)
* Circle group (uncountably infinite analogue)

Notes

References

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{{DEFAULTSORT:Prufer Group
Abelian group theory
Infinite group theory
P-groups