Prüfer group

Short description: Mathematical term in group theory

The Prüfer 2-group with presentation ⟨gn: gn+12 = gn, g12 = e⟩, illustrated as a subgroup of the unit circle in the complex plane Algebraic structure → Ring theory
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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.

## Contents

* 1 Constructions
* 2 Properties
* 3 See also
* 4 Notes
* 5 References

## Constructions

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:

ℤ(p∞)={exp⁡(2πim/pn)∣0≤m<pn,n∈ℤ+}={z∈ℂ∣z(pn)=1 for some n∈ℤ+}.

The group operation here is the multiplication of complex numbers.

There is a presentation

ℤ(p∞)=⟨g1,g2,g3,…∣g1p=1,g2p=g1,g3p=g2,…⟩.

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:

ℤ(p∞)=ℤ[1/p]/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/pnℤ and the embedding ℤ/pnℤ->ℤ/pn+1ℤ induced by multiplication by p. The direct limit of this system is ℤ(p∞):

ℤ(p∞)=lim->ℤ/pnℤ.

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

We can also write

ℤ(p∞)=ℚp/ℤp

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(p∞) is:

0⊊(1pℤ)/ℤ⊊(1p2ℤ)/ℤ⊊(1p3ℤ)/ℤ⊊⋯⊊ℤ(p∞)

Here, each (1pnℤ)/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 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.[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 (cardinal) 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]

## 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/2b. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1.
* Cyclic group (finite analogue)
* Circle group (uncountably infinite analogue)

## Notes

1. ↑ See Vil'yams (2001)
2. ↑ See Kaplansky (1965)
3. ↑ See also Jacobson (2009), p. 102, ex. 2.
4. ↑ See Vil'yams (2001)
5. ↑ D. L. Armacost and W. L. Armacost,"On p-thetic groups", Pacific J. Math., 41, no. 2 (1972), 295–301

## References

* Jacobson, Nathan (2009). Basic algebra. 2 (2nd ed.). Dover. ISBN 978-0-486-47187-7.
* Pierre Antoine Grillet (2007). Abstract algebra. Springer. ISBN 978-0-387-71567-4.
* Kaplansky, Irving (1965). Infinite Abelian Groups. University of Michigan Press.
* Hazewinkel, Michiel, ed. (2001), "Quasi-cyclic group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Q/q076440

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