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Prüfer group

Contents

Definition

For a prime number pp, the Prüfer pp-group is defined uniquely up to isomorphism as the group where every element has exactly pp p thp^{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 [1/p]/\mathbb{Z}[1/p]/\mathbb{Z}, the colimit of the sequence of inclusions

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

Properties

The Prüfer pp-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.

Revised on January 31, 2012 20:30:50 by Urs Schreiber (82.169.65.155)