quasicyclic group

Let $p$ be a prime number. The $p$-quasicyclic group (or Prüfer $p$-group, or $p^{\mathrm{\infty }}$ 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 $\mathbb{Z}(p^{\mathrm{\infty }})$. Other notations in use include $\mathbb{Z}[p^{\mathrm{\infty }}]$, $\mathbb{Z}/p^{\mathrm{\infty }}\mathbb{Z}$, ${\mathbb{Z}}_{p^{\mathrm{\infty }}}$ and $C_{p^{\mathrm{\infty }}}$.

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

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

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

• $\mathbb{Z}(p^{\mathrm{\infty }})$ 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 $\mathbb{Z}(p^{\mathrm{\infty }})$ 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 $\mathbb{Z}(p^{\mathrm{\infty }})$.

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