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
):
-
•
is the group of all -th complex roots of , for .
-
•
is the injective hull of (viewing abelian groups
as -modules (http://planetmath.org/Module)).
-
•
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![]() |