quasicyclic group

Let p be a prime number. The p-quasicyclic group (or Prüfer p-group, or p group) is the p-primary componentPlanetmathPlanetmath of /, that is, the unique maximal p-subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/PGroup4) of /. Any group (http://planetmath.org/Group) isomorphicPlanetmathPlanetmathPlanetmath to this will also be called a p-quasicyclic group.

The p-quasicyclic group will be denoted by (p). Other notations in use include [p], /p, p and Cp.

(p) may also be defined in a number of other (equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) ways (again, up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath):

  • (p) is the group of all pn-th complex roots of 1, for n.

  • (p) is the injective hull of /p (viewing abelian groupsMathworldPlanetmath as -modules (http://planetmath.org/Module)).

  • (p) is the direct limitMathworldPlanetmath of the groups /pn.

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

The subgroup (http://planetmath.org/Subgroup) structureMathworldPlanetmath of (p) is particularly simple: all proper subgroupsMathworldPlanetmath are finite and cyclic, and there is exactly one of order pn 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 infiniteMathworldPlanetmath solvable groupsMathworldPlanetmath 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 (p).

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-groupMathworldPlanetmath