Prüfer group

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The Prüfer 2-group. <gn: gn+12 = gn, g12 = e>

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 pth roots. The group is named after Heinz Prüfer. It is a countable abelian group which helps taxonomize infinite abelian groups.

The Prüfer p-group may be represented as a subgroup of the circle group, U(1), as the set of pnth roots of unity as n ranges over all non-negative integers:

\mathbf{Z}(p^\infty)=\{\exp(2\pi i m/p^n) \mid m\in \mathbf{Z}^+,\,n\in \mathbf{Z}^+\}.\;

Alternatively, the Prüfer p-group may be seen as the Sylow p-subgroup of Q/Z, consisting of those elements whose order is a power of p:

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

or equivalently \mathbf{Z}(p^\infty)=\mathbf{Q}_p/\mathbf{Z}_p.

There is a presentation

\mathbf{Z}(p^\infty) = \langle\, x_1, x_2, x_3, \ldots \mid x_1^p = 1, x_2^p = x_1, x_3^p = x_2, \dots\,\rangle.

The Prüfer p-group is the unique infinite p-group which is locally cyclic (every finite set of elements generates a cyclic group).

The Prüfer p-group is divisible.

In the language of universal algebra, an abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic p-group or isomorphic to a Prüfer group.

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.[1]

The Prüfer p-groups for all primes p are the only infinite groups whose subgroups are totally ordered by inclusion. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup.

0 \subset \mathbf{Z}/p \subset \mathbf{Z}/p^2 \subset \mathbf{Z}/p^3 \subset \cdots \subset \mathbf{Z}(p^\infty)

This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups.

As an Abelian group (that is, as a \mathbb{Z},-module), it is Artinian but not Noetherian.[2][3] It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian ring is Noetherian).

See also[edit]

  • 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.


  1. ^ D. L. Armacost and W. L. Armacost,"On p-thetic groups", Pacific J. Math., 41, no. 2 (1972), 295–301
  2. ^ Subgroups of an Abelian group are Abelian, and so are submodules of \mathbb{Z}-modules.
  3. ^ See also Jacobson (2009), p. 102, ex. 2.