quasicyclic group
Let be a prime number. The -quasicyclic group (or Prüfer -group, or group) is the -primary component of , that is, the unique maximal -subgroup (http://planetmath.org/PGroup4) of . Any group (http://planetmath.org/Group) isomorphic to this will also be called a -quasicyclic group.
The -quasicyclic group will be denoted by . Other notations in use include , , and .
may also be defined in a number of other (equivalent) ways (again, up to isomorphism):
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is the group of all -th complex roots of , for .
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is the injective hull of (viewing abelian groups as -modules (http://planetmath.org/Module)).
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is the direct limit of the groups .
A quasicyclic group (or Prüfer group) is a group that is -quasicyclic for some prime .
The subgroup (http://planetmath.org/Subgroup) structure of 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 (http://planetmath.org/DivisibleGroup) and co-Hopfian.
Every infinite locally cyclic -group is isomorphic to .
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 |