In mathematics, specifically ingroup theory, thePrüferp-group or thep-quasicyclic group orp^∞-group,Z(p^∞), for aprime numberp is the uniquep-group in which every element hasp differentp-th roots.

The Prüferp-groups arecountableabelian groups that are important in the classification of infinite abelian groups: they (along with the group ofrational numbers) form the smallest building blocks of alldivisible groups.

The groups are named afterHeinz Prüfer, a German mathematician of the early 20th century.

The Prüferp-group may be identified with the subgroup of thecircle group, U(1), consisting of allpⁿ-throots of unity asn ranges over all non-negative integers:

The group operation here is the multiplication ofcomplex numbers.

There is apresentation

${\displaystyle \mathbf{Z}(p^{\mathrm{\infty }})=⟨g_1,g_2,g_3,\dots \mid g_1^p=1,g_2^p=g_1,g_3^p=g_2,\dots ⟩.}$

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

Alternatively and equivalently, the Prüferp-group may be defined as theSylowp-subgroup of thequotient groupQ/Z, consisting of those elements whose order is a power ofp:

${\displaystyle \mathbf{Z}(p^{\mathrm{\infty }})=\mathbf{Z}[1/p]/\mathbf{Z}}$

(whereZ[1/p] denotes the group of all rational numbers whose denominator is a power ofp, using addition of rational numbers as group operation).

For each natural numbern, consider thequotient groupZ/pⁿZ and the embeddingZ/pⁿZ →Z/pⁿ⁺¹Z induced by multiplication byp. Thedirect limit of this system isZ(p^∞):

${\displaystyle \mathbf{Z}(p^{\mathrm{\infty }})=\underrightarrow{\lim }\mathbf{Z}/p^n\mathbf{Z}.}$

If we perform the direct limit in thecategory oftopological groups, then we need to impose a topology on each of the${\displaystyle \mathbf{Z}/p^n\mathbf{Z}}$, and take thefinal topology on${\displaystyle \mathbf{Z}(p^{\mathrm{\infty }})}$. If we wish for${\displaystyle \mathbf{Z}(p^{\mathrm{\infty }})}$ to beHausdorff, we must impose thediscrete topology on each of the${\displaystyle \mathbf{Z}/p^n\mathbf{Z}}$, resulting in${\displaystyle \mathbf{Z}(p^{\mathrm{\infty }})}$ to have the discrete topology.

We can also write

${\displaystyle \mathbf{Z}(p^{\mathrm{\infty }})={\mathbf{Q}}_p/{\mathbf{Z}}_p}$

whereQ_p denotes the additive group ofp-adic numbers andZ_p is the subgroup ofp-adic integers.

The complete list of subgroups of the Prüferp-groupZ(p^∞) =Z[1/p]/Z is:

${\displaystyle 0⊊\left(\frac{1}{p}\mathbf{Z}\right)/\mathbf{Z}⊊\left(\frac{1}{p^2}\mathbf{Z}\right)/\mathbf{Z}⊊\left(\frac{1}{p^3}\mathbf{Z}\right)/\mathbf{Z}⊊\cdots ⊊\mathbf{Z}(p^{\mathrm{\infty }})}$

Here, each${\displaystyle \left(\frac{1}{p^n}\mathbf{Z}\right)/\mathbf{Z}}$ is a cyclic subgroup ofZ(p^∞) withpⁿ elements; it contains precisely those elements ofZ(p^∞) whoseorder dividespⁿ and corresponds to the set ofpⁿ-th roots of unity.

The Prüferp-groups are the only infinite groups whose subgroups aretotally ordered by inclusion. This sequence of inclusions expresses the Prüferp-group as thedirect limit of its finite subgroups. As there is nomaximal subgroup of a Prüferp-group, it is its ownFrattini subgroup.

Given this list of subgroups, it is clear that the Prüferp-groups areindecomposable (cannot be written as adirect sum of proper subgroups). More is true: the Prüferp-groups aresubdirectly irreducible. An abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclicp-group or to a Prüfer group.

The Prüferp-group is the unique infinitep-group that islocally cyclic (every finite set of elements generates a cyclic group). As seen above, all proper subgroups ofZ(p^∞) are finite. The Prüferp-groups are the only infinite abelian groups with this property.^[1]

The Prüferp-groups aredivisible. 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 thedirect sum of a (possibly infinite) number of copies ofQ and (possibly infinite) numbers of copies ofZ(p^∞) for every primep. The (cardinal) numbers of copies ofQ andZ(p^∞) that are used in this direct sum determine the divisible group up to isomorphism.^[2]

As an abelian group (that is, as aZ-module),Z(p^∞) isArtinian but notNoetherian.^[3] It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas everyArtinianring is Noetherian).

Theendomorphism ring ofZ(p^∞) is isomorphic to the ring ofp-adic integersZ_p.^[4]

In the theory oflocally compact topological groups the Prüferp-group (endowed with thediscrete topology) is thePontryagin dual of the compact group ofp-adic integers, and the group ofp-adic integers is the Pontryagin dual of the Prüferp-group.^[5]

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

• Jacobson, Nathan (2009).Basic algebra. Vol. 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.
• N.N. Vil'yams (2001) [1994],"Quasi-cyclic group",Encyclopedia of Mathematics,EMS Press

• Abelian group theory
• Infinite group theory
• P-groups

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