# quasicyclic group

Let $p$ be a prime number. The $p$-quasicyclic group (or Prüfer $p$-group, or $p^{\infty}$ group) is the $p$-primary component of $\mathbb{Q}/\mathbb{Z}$, that is, the unique maximal $p$-subgroup (http://planetmath.org/PGroup4) of $\mathbb{Q}/\mathbb{Z}$. Any group (http://planetmath.org/Group) isomorphic to this will also be called a $p$-quasicyclic group.

The $p$-quasicyclic group will be denoted by $\mathbb{Z}(p^{\infty})$. Other notations in use include $\mathbb{Z}[p^{\infty}]$, $\mathbb{Z}/p^{\infty}\mathbb{Z}$, $\mathbb{Z}_{p^{\infty}}$ and $C_{p^{\infty}}$.

$\mathbb{Z}(p^{\infty})$ may also be defined in a number of other (equivalent) ways (again, up to isomorphism):

• $\mathbb{Z}(p^{\infty})$ is the group of all $p^{n}$-th complex roots of $1$, for $n\in\mathbb{N}$.

• $\mathbb{Z}(p^{\infty})$ is the injective hull of $\mathbb{Z}/p\mathbb{Z}$ (viewing abelian groups as $\mathbb{Z}$-modules (http://planetmath.org/Module)).

• $\mathbb{Z}(p^{\infty})$ is the direct limit of the groups $\mathbb{Z}/p^{n}\mathbb{Z}$.

A quasicyclic group (or Prüfer group) is a group that is $p$-quasicyclic for some prime $p$.

The subgroup (http://planetmath.org/Subgroup) structure of $\mathbb{Z}(p^{\infty})$ is particularly simple: all proper subgroups are finite and cyclic, and there is exactly one of order $p^{n}$ for each non-negative integer $n$. In particular, this means that the subgroups are linearly ordered by inclusion, and all subgroups are fully invariant. The quasicyclic groups are the only infinite groups with a linearly ordered subgroup lattice. They are also the only infinite solvable groups whose proper subgroups are all finite.

Quasicyclic groups are locally cyclic, divisible (http://planetmath.org/DivisibleGroup) and co-Hopfian.

Every infinite locally cyclic $p$-group is isomorphic to $\mathbb{Z}(p^{\infty})$.

 Title quasicyclic group Canonical name QuasicyclicGroup Date of creation 2013-03-22 15:35:22 Last modified on 2013-03-22 15:35:22 Owner yark (2760) Last modified by yark (2760) Numerical id 19 Author yark (2760) Entry type Definition Classification msc 20F50 Classification msc 20K10 Synonym quasi-cyclic group Synonym Prüfer group Defines quasicyclic Defines quasi-cyclic Defines Prüfer p-group