Prüfer group

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups

Definition

For $p$ a prime number, the Prüfer group is the p-group whose elements all have $p$ different $p$th roots.

It can be realized as the subgroup of $\mathbb {C}$ as follows:

$G=\{z\in \mathbb {C} :z^{p^{n}}=1{\text{ for some }}n\}$

Properties

It is a countably infinite group.

It is an abelian group, since it is a subgroup of the multiplicative group of non-zero complex numbers, which is abelian.

Particular cases

$p$	Cyclic group of order $n$
2	Prüfer 2-group
3	Prüfer 3-group
5	Prüfer 5-group
7	Prüfer 7-group
11	Prüfer 11-group
13	Prüfer 13-group
17	Prüfer 17-group
19	Prüfer 19-group
23	Prüfer 23-group