# Prüfer group

Algebraic structure → Ring theoryRing theory |
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In mathematics, specifically in group theory, the **Prüfer p-group** or the

**or**

*p*-quasicyclic group**-group,**

*p*^{∞}**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:

- [math]\displaystyle{ \mathbf{Z}(p^\infty)=\{\exp(2\pi i m/p^n) \mid 0 \leq m \lt p^n,\,n\in \mathbf{Z}^+\} = \{z\in\mathbf{C} \mid z^{(p^n)}=1 \text{ for some } n\in \mathbf{Z}^+\}.\; }[/math]

The group operation here is the multiplication of complex numbers.

There is a presentation

- [math]\displaystyle{ \mathbf{Z}(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. }[/math]

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*:

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

(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**/*p*^{n}**Z** and the embedding **Z**/*p*^{n}**Z** → **Z**/*p*^{n+1}**Z** induced by multiplication by *p*. The direct limit of this system is **Z**(*p*^{∞}):

- [math]\displaystyle{ \mathbf{Z}(p^\infty) = \varinjlim \mathbf{Z}/p^n \mathbf{Z} . }[/math]

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

We can also write

- [math]\displaystyle{ \mathbf{Z}(p^\infty)=\mathbf{Q}_p/\mathbf{Z}_p }[/math]

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**[1/*p*]/**Z** is:

- [math]\displaystyle{ 0 \subsetneq \left({1 \over p}\mathbf{Z}\right)/\mathbf{Z} \subsetneq \left({1 \over p^2}\mathbf{Z}\right)/\mathbf{Z} \subsetneq \left({1 \over p^3}\mathbf{Z}\right)/\mathbf{Z} \subsetneq \cdots \subsetneq \mathbf{Z}(p^\infty) }[/math]

Here, each [math]\displaystyle{ \left({1 \over p^n}\mathbf{Z}\right)/\mathbf{Z} }[/math] 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 *p ^{n}*-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 **Z**_{p}.^{[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*/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

- 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

Original source: https://en.wikipedia.org/wiki/Prüfer group.
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