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Definition

For a prime number $p$, the Prüfer $p$-group is defined uniquely up to isomorphism as the group $p$-group where every element has exactly $p$ $p^{th}$ 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 (for more on this see as Pontryagin duality for torsion abelian groups).

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

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

As such, it is the initial algebra of the functor $F: \mathbb{Z} \darr LCHAb \to \mathbb{Z} \darr LCHAb$ that pushes out along multiplication by $p: \mathbb{Z} \to \mathbb{Z}$.

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.

Related concepts

• Q/Z

• localization of abelian groups