PlanetMath: quasicyclic group

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quasicyclic group

(Definition)

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

The -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 -th complex roots of , 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).
$\mathbb{Z}(p^\infty)$ is the direct limit of the groups $\mathbb{Z}/p^n\mathbb{Z}$ .

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

The subgroup structure of $\mathbb{Z}(p^\infty)$ is particularly simple: all proper subgroups are finite and cyclic, and there is exactly one of order for each non-negative integer . 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 and co-Hopfian.

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

"quasicyclic group" is owned by yark.

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Other names:

quasi-cyclic group, Prüfer group

Also defines:

quasicyclic, quasi-cyclic, Prüfer p-group

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Cross-references: co-Hopfian, locally cyclic, solvable groups, subgroup lattice, infinite, fully invariant, inclusion, linearly ordered, integer, order, cyclic, finite, proper subgroups, direct limit, abelian groups, injective hull, complex roots, isomorphism, number, isomorphic, prime number
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This is version 16 of quasicyclic group, born on 2005-11-25, modified 2006-03-23.
Object id is 7500, canonical name is QuasicyclicGroup.
Accessed 5126 times total.

Classification:

AMS MSC:	20K10 (Group theory and generalizations :: Abelian groups :: Torsion groups, primary groups and generalized primary groups)
	20F50 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Periodic groups; locally finite groups)

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