nLab
Prüfer group

Definition

For a prime number $p$, the Prüfer $p$-group is defined uniquely up to isomorphism as the group where every element has exactly $p$ $p^{\mathrm{th}}$ roots. It is a divisible abelian group which can be described in several ways, for example:

• It is the discrete group that is Pontryagin dual to the compact topological group of p-adic integers.

• It is $\mathbb{Z}[1/p]/\mathbb{Z}$, the colimit of the sequence of inclusions

  $$0\hookrightarrow \mathbb{Z}/(p)\hookrightarrow \dots \hookrightarrow \mathbb{Z}/(p^n)\hookrightarrow \mathbb{Z}/(p^{n+1})\hookrightarrow \dots$$0 \hookrightarrow \mathbb{Z}/(p) \hookrightarrow \ldots \hookrightarrow \mathbb{Z}/(p^n) \hookrightarrow \mathbb{Z}/(p^{n+1}) \hookrightarrow \ldots

Properties

The Prüfer $p$-groups are the only infinite groups whose subgroups are totally ordered by inclusion. They are often useful as counterexamples in algebra; for example, a Prüfer group is an Artinian but not a Noetherian $\mathbb{Z}$-module.