nLab Prüfer group




For a prime number pp, the Prüfer pp-group is defined uniquely up to isomorphism as the p p -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:

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


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.

Last revised on May 22, 2024 at 23:07:54. See the history of this page for a list of all contributions to it.