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^{}):

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

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

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$\mathbb{Z}({p}^{\mathrm{\infty}})$ 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 nonnegative 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 coHopfian.
Every infinite locally cyclic $p$group is isomorphic to $\mathbb{Z}({p}^{\mathrm{\infty}})$.
Title  quasicyclic group 
Canonical name  QuasicyclicGroup 
Date of creation  20130322 15:35:22 
Last modified on  20130322 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  quasicyclic group 
Synonym  Prüfer group 
Defines  quasicyclic 
Defines  quasicyclic 
Defines  Prüfer pgroup^{} 